magnetic field travel speed

11.2 Magnetic Fields and Lines

Learning objectives.

By the end of this section, you will be able to:

• Define the magnetic field based on a moving charge experiencing a force
• Apply the right-hand rule to determine the direction of a magnetic force based on the motion of a charge in a magnetic field
• Sketch magnetic field lines to understand which way the magnetic field points and how strong it is in a region of space

We have outlined the properties of magnets, described how they behave, and listed some of the applications of magnetic properties. Even though there are no such things as isolated magnetic charges, we can still define the attraction and repulsion of magnets as based on a field. In this section, we define the magnetic field, determine its direction based on the right-hand rule, and discuss how to draw magnetic field lines.

Defining the Magnetic Field

A magnetic field is defined by the force that a charged particle experiences moving in this field, after we account for the gravitational and any additional electric forces possible on the charge. The magnitude of this force is proportional to the amount of charge q , the speed of the charged particle v , and the magnitude of the applied magnetic field. The direction of this force is perpendicular to both the direction of the moving charged particle and the direction of the applied magnetic field. Based on these observations, we define the magnetic field strength B based on the magnetic force F → F → on a charge q moving at velocity v → v → as the cross product of the velocity and magnetic field, that is,

In fact, this is how we define the magnetic field B → B → —in terms of the force on a charged particle moving in a magnetic field. The magnitude of the force is determined from the definition of the cross product as it relates to the magnitudes of each of the vectors. In other words, the magnitude of the force satisfies

where θ is the angle between the velocity and the magnetic field.

The SI unit for magnetic field strength B is called the tesla (T) after the eccentric but brilliant inventor Nikola Tesla (1856–1943), where

A smaller unit, called the gauss (G), where 1 G = 10 −4 T , 1 G = 10 −4 T , is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. Earth’s magnetic field on its surface is only about 5 × 10 −5 T , 5 × 10 −5 T , or 0.5 G.

Problem-Solving Strategy

Direction of the magnetic field by the right-hand rule.

The direction of the magnetic force F → F → is perpendicular to the plane formed by v → v → and B → , B → , as determined by the right-hand rule-1 (or RHR-1), which is illustrated in Figure 11.4 .

• Orient your right hand so that your fingers curl in the plane defined by the velocity and magnetic field vectors.
• Using your right hand, sweep from the velocity toward the magnetic field with your fingers through the smallest angle possible.
• The magnetic force is directed where your thumb is pointing.
• If the charge was negative, reverse the direction found by these steps.

Interactive

Visit this website for additional practice with the direction of magnetic fields.

There is no magnetic force on static charges. However, there is a magnetic force on charges moving at an angle to a magnetic field. When charges are stationary, their electric fields do not affect magnets. However, when charges move, they produce magnetic fields that exert forces on other magnets. When there is relative motion, a connection between electric and magnetic forces emerges—each affects the other.

Example 11.1

An alpha-particle moving in a magnetic field.

• First, to determine the direction, start with your fingers pointing in the positive x -direction. Sweep your fingers upward in the direction of magnetic field. Your thumb should point in the negative y -direction. This should match the mathematical answer. To calculate the force, we use the given charge, velocity, and magnetic field and the definition of the magnetic force in cross-product form to calculate: F → = q v → × B → = ( 3.2 × 10 −19 C ) ( 5.0 × 10 4 m/s i ^ ) × ( 1.5 T k ^ ) = −2.4 × 10 −14 N j ^ . F → = q v → × B → = ( 3.2 × 10 −19 C ) ( 5.0 × 10 4 m/s i ^ ) × ( 1.5 T k ^ ) = −2.4 × 10 −14 N j ^ .
• First, to determine the directionality, start with your fingers pointing in the negative y -direction. Sweep your fingers upward in the direction of magnetic field as in the previous problem. Your thumb should be open in the negative x -direction. This should match the mathematical answer. To calculate the force, we use the given charge, velocity, and magnetic field and the definition of the magnetic force in cross-product form to calculate: F → = q v → × B → = ( 3.2 × 10 −19 C ) ( −5.0 × 10 4 m/s j ^ ) × ( 1.5 T k ^ ) = −2.4 × 10 −14 N i ^ . F → = q v → × B → = ( 3.2 × 10 −19 C ) ( −5.0 × 10 4 m/s j ^ ) × ( 1.5 T k ^ ) = −2.4 × 10 −14 N i ^ . An alternative approach is to use Equation 11.2 to find the magnitude of the force. This applies for both parts (a) and (b). Since the velocity is perpendicular to the magnetic field, the angle between them is 90 degrees. Therefore, the magnitude of the force is: F = q v B sin θ = ( 3.2 × 10 −19 C ) ( 5.0 × 10 4 m / s ) ( 1.5 T ) sin ( 90 ° ) = 2.4 × 10 −14 N. F = q v B sin θ = ( 3.2 × 10 −19 C ) ( 5.0 × 10 4 m / s ) ( 1.5 T ) sin ( 90 ° ) = 2.4 × 10 −14 N.
• Since the velocity and magnetic field are parallel to each other, there is no orientation of your hand that will result in a force direction. Therefore, the force on this moving charge is zero. This is confirmed by the cross product. When you cross two vectors pointing in the same direction, the result is equal to zero.
• First, to determine the direction, your fingers could point in any orientation; however, you must sweep your fingers upward in the direction of the magnetic field. As you rotate your hand, notice that the thumb can point in any x - or y -direction possible, but not in the z -direction. This should match the mathematical answer. To calculate the force, we use the given charge, velocity, and magnetic field and the definition of the magnetic force in cross-product form to calculate: F → = q v → × B → = ( 3.2 × 10 −19 C ) ( ( 2.0 i ^ − 3.0 j ^ + 1.0 k ^ ) × 10 4 m/s ) × ( 1.5 T k ^ ) = ( −14.4 i ^ − 9.6 j ^ ) × 10 −15 N. F → = q v → × B → = ( 3.2 × 10 −19 C ) ( ( 2.0 i ^ − 3.0 j ^ + 1.0 k ^ ) × 10 4 m/s ) × ( 1.5 T k ^ ) = ( −14.4 i ^ − 9.6 j ^ ) × 10 −15 N. This solution can be rewritten in terms of a magnitude and angle in the xy -plane: | F → | = F x 2 + F y 2 = ( −14.4 ) 2 + ( −9.6 ) 2 × 10 −15 N = 1.7 × 10 −14 N θ = tan −1 ( F y F x ) = tan −1 ( −9.6 × 10 −15 N −14.4 × 10 −15 N ) = 34 ° . | F → | = F x 2 + F y 2 = ( −14.4 ) 2 + ( −9.6 ) 2 × 10 −15 N = 1.7 × 10 −14 N θ = tan −1 ( F y F x ) = tan −1 ( −9.6 × 10 −15 N −14.4 × 10 −15 N ) = 34 ° . The magnitude of the force can also be calculated using Equation 11.2 . The velocity in this question, however, has three components. The z -component of the velocity can be neglected, because it is parallel to the magnetic field and therefore generates no force. The magnitude of the velocity is calculated from the x - and y -components. The angle between the velocity in the xy -plane and the magnetic field in the z -plane is 90 degrees. Therefore, the force is calculated to be: | v → | = ( 2 ) 2 + ( −3 ) 2 × 10 4 m s = 3.6 × 10 4 m s F = q v B sin θ = ( 3.2 × 10 −19 C ) ( 3.6 × 10 4 m/s ) ( 1.5 T ) sin ( 90 ° ) = 1.7 × 10 −14 N. | v → | = ( 2 ) 2 + ( −3 ) 2 × 10 4 m s = 3.6 × 10 4 m s F = q v B sin θ = ( 3.2 × 10 −19 C ) ( 3.6 × 10 4 m/s ) ( 1.5 T ) sin ( 90 ° ) = 1.7 × 10 −14 N. This is the same magnitude of force calculated by unit vectors.

Significance

Check your understanding 11.1.

Repeat the previous problem with the magnetic field in the x -direction rather than in the z -direction. Check your answers with RHR-1.

Representing Magnetic Fields

The representation of magnetic fields by magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. As shown in Figure 11.6 , each of these lines forms a closed loop, even if not shown by the constraints of the space available for the figure. The field lines emerge from the north pole (N), loop around to the south pole (S), and continue through the bar magnet back to the north pole.

Magnetic field lines have several hard-and-fast rules:

• The direction of the magnetic field is tangent to the field line at any point in space. A small compass will point in the direction of the field line.
• The strength of the field is proportional to the closeness of the lines. It is exactly proportional to the number of lines per unit area perpendicular to the lines (called the areal density).
• Magnetic field lines can never cross, meaning that the field is unique at any point in space.
• Magnetic field lines are continuous, forming closed loops without a beginning or end. They are directed from the north pole to the south pole.

The last property is related to the fact that the north and south poles cannot be separated. It is a distinct difference from electric field lines, which generally begin on positive charges and end on negative charges or at infinity. If isolated magnetic charges (referred to as magnetic monopoles ) existed, then magnetic field lines would begin and end on them.

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Physics library

Course: physics library   >   unit 13.

• Magnetic field created by a current carrying wire
• What are magnetic fields?
• Magnetic force between two currents going in the same direction

3 Ways Fundamental Particles Travel at (Nearly) the Speed of Light

Light-speed travel is a staple of science fiction in space. No "Star Wars" movie seems complete until the Millennium Falcon (or a rival ship) uses its hyperdrive. And many "Star Trek" fans enjoy talking about the relative star-system-jumping speeds of the USS Enterprise, against the speeds of other Federation ships.

But in real life, physics gets in the way. Einstein's theory of special relativity essentially puts a speed limit on cosmic travel; as far as we can tell, nothing goes faster than the speed of light. Worse, any object that has mass tends to get more and more massive — dragging down the object's velocity — as it approaches light speed. So as far as we know, only small particles can get anywhere near the speed of light.

One hundred years ago, on May 29, 1919, scientists performed measurements of a solar eclipse that confirmed Einstein's work. To celebrate, NASA offered three ways that particles can accelerate to amazing speed in a new statement .

Related: Why Don't We Have a 'Star Wars' Hyperdrive Yet?  

Electromagnetic fields

The sun is a wacky environment to study physics, because it is so extreme compared to Earth. It's also a real-life laboratory showing how nuclear reactions happen. It also is an example of an environment with electromagnetic fields — which, as NASA points out, is the same force that stops magnets from falling off your fridge.

Magnetic fields and electric fields work together to accelerate particles with an electric charge. This charge allows electromagnetic fields to push particles along — sometimes at speeds approaching the speed of light.

We can even simulate this process on Earth. Huge particle accelerators (like at the Department of Energy's Fermi National Accelerator Laboratory, or at the European Organization for Nuclear Research's Large Hadron Collider ) create pulsed electromagnetic fields. These fields accelerate charged particles close to the speed of light. Next, scientists often crash these particles together to see what particles and energy are released. 

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In fractions of a second after these collisions, we can quickly observe elementary particles that were around in the first few seconds after the universe was formed. (That event, called the Big Bang , happened about 13.8 billion years ago.)

Magnetic explosions

The sun is also host to phenomena called solar flares . Dancing above the sun's surface is a tangle of magnetic fields. At times, these fields intersect and snap, sending plumes of solar material off the surface — and, sometimes, charged particles along with it.

"When the tension between the crossed lines becomes too great, the lines explosively snap and realign in a process known as magnetic reconnection," NASA officials said in the statement. "The rapid change in a region's magnetic field creates electric fields, which causes all the attendant charged particles to be flung away at high speeds."

Particles streaming off the sun may accelerate close to the speed of light, thrown from the sun thanks to magnetic reconnection. One example of such objects is the solar wind , the constant stream of charged particles the sun emits into the solar system. (There may be other factors speeding these particles as well, such as wave-particle interactions — which is explained in the next section of this article.) 

Magnetic reconnection also likely happens at large planets, such as Jupiter and Saturn. Closer to home, NASA studies magnetic reconnection near Earth using the Magnetospheric Multiscale mission , which measures our planet's magnetic field using four spacecraft. The results may be useful to better understand how particles accelerate all over the universe, NASA officials said.

Wave-particle interactions

Particles can also careen at high speeds when electromagnetic waves collide; that phenomenon is more technically called wave-particle interactions.

"When electromagnetic waves collide, their fields can become compressed. Charged particles bouncing back and forth between the waves can gain energy similar to a ball bouncing between two merging walls," NASA officials said.

These interactions take place all over the universe. Near Earth, NASA missions such as the Van Allen probes are watching wave-particle interactions to better predict particle movements — and protect electronics on satellites. That's because high-speed particles can damage these delicate spacecraft parts.

Supernovas, or star explosions, may also play a role in more far-away interactions. Researchers have theorized that after a star explodes, it creates a blast wave — a shell of hot, dense compressed gas — that zooms away from the stellar core at high speed. These bubbles are full of charged particles and magnetic fields, creating a likely environment for wave-particle interactions. This process may eject high-energy cosmic rays — which consist of particles —  at velocities close to the speed of light. 

• Space Magnetism May Hold Secret to Fusion Power
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Follow Elizabeth Howell on Twitter @howellspace . Follow us on Twitter @Spacedotcom and on Facebook .

Join our Space Forums to keep talking space on the latest missions, night sky and more! And if you have a news tip, correction or comment, let us know at: [email protected].

Elizabeth Howell (she/her), Ph.D., is a staff writer in the spaceflight channel since 2022 covering diversity, education and gaming as well. She was contributing writer for Space.com for 10 years before joining full-time. Elizabeth's reporting includes multiple exclusives with the White House and Office of the Vice-President of the United States, an exclusive conversation with aspiring space tourist (and NSYNC bassist) Lance Bass, speaking several times with the International Space Station, witnessing five human spaceflight launches on two continents, flying parabolic, working inside a spacesuit, and participating in a simulated Mars mission. Her latest book, " Why Am I Taller ?", is co-written with astronaut Dave Williams. Elizabeth holds a Ph.D. and M.Sc. in Space Studies from the University of North Dakota, a Bachelor of Journalism from Canada's Carleton University and a Bachelor of History from Canada's Athabasca University. Elizabeth is also a post-secondary instructor in communications and science at several institutions since 2015; her experience includes developing and teaching an astronomy course at Canada's Algonquin College (with Indigenous content as well) to more than 1,000 students since 2020. Elizabeth first got interested in space after watching the movie Apollo 13 in 1996, and still wants to be an astronaut someday. Mastodon: https://qoto.org/@howellspace

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The National MagLab is funded by the National Science Foundation and the State of Florida.

James Clerk Maxwell

James Clerk Maxwell was one of the most influential scientists of the nineteenth century.

His theoretical work on electromagnetism and light largely determined the direction that physics would take in the early 20th century. Indeed, according to Albert Einstein, “One scientific epoch ended and another began with James Clerk Maxwell.”

When he was born on June 13, 1831 in Edinburgh, Scotland, the scientist-to-be was only known as James Clerk, but the surname of Maxwell was added to his appellation when his father, an attorney, inherited an estate from ancestors with that name. Maxwell was an only child, and his mother died from cancer when he was 8 years old. A tutor was at first engaged to educate him, but then, in 1841, he was enrolled at the Edinburgh Academy. His interests were wide ranging, and at the age of 14 his first paper was published. The subject was geometry, and his skill in this and other mathematical spheres would aid him in his many scientific endeavors. Maxwell began studying at the University of Edinburgh in 1847 and published two more papers while still a teenager.

Maxwell transferred in 1850 to Cambridge University, where he was an exemplary student and won various awards, including the Smith’s Prize. He graduated in 1854 and accepted a Trinity College fellowship. As a fellow, Maxwell began research into two topics that he would investigate throughout his life: color and magnetism. This work resulted in the publication of two papers in 1855, “Experiments on Colour, as perceived by the Eye, with remarks on Colour-blindness” and “On Faraday’s Lines of Force.” That same year, Maxwell was elected a Fellow of the Royal Society of Edinburgh, and the following year he received an appointment as professor of natural philosophy at the University of Aberdeen’s Marischal College. His father, with whom he had been very close, died shortly before the appointment was made, and Maxwell inherited the family estate. In 1858, he married Katherine Mary Dewar, whom he met through a colleague at the college.

During his years at Aberdeen, Maxwell carried out research into a number of areas, but became particularly occupied with the nature of Saturn's rings, the subject of the Adams Prize of 1857. Deciding to compete for the prize, he spent two years attempting to find a way to accurately determine the composition of the rings. Eventually he theorized, using purely mathematical reasoning, that the rings could not be stable if they were comprised of a homogenous solid, leading him to conclude that the rings must be made of an unknown number of unconnected particles. Maxwell’s theory, finally proven a century later when the Voyager space probes were sent to Saturn, won him the prestigious prize. This research lead to more general inquiries into heat and the kinetics of gases. In 1859, Maxwell developed a statistical description of velocity distribution among the molecules comprising a gas, which would eventually be expanded into the Maxwell-Boltzmann distribution law .

Maxwell accepted a professorial position at King’s College, London in 1860. The five years he was associated with the institution are generally regarded as his most scientifically profitable. At this time, he applied his earlier studies of color vision and optics to photography, producing the world’s first color photograph in 1861. To do this he developed a trichromatic process, in which the same subject was photographed through red, blue and green color filters, and the three resulting images were combined into one. Also during his King’s College years, Maxwell continued his work with gases, which would culminate in his important treatise “On the Dynamical Theory of Gases” in 1867, and made groundbreaking advances in the area of electromagnetism.

It is for his electromagnetic theory that Maxwell is most commonly credited with fundamentally changing the course of physics. To arrive at his theory, Maxwell borrowed and extended ideas previously developed by several other scientists, including Michael Faraday , William Thomson (Lord Kelvin), and Carl Friedrich Gauss , among others. From his attempt to translate the experimental findings of Faraday into the language of mathematics, Maxwell arrived at a set of equations that comprehensively describe the production and the relationships between electric fields and magnetic fields. Based on the equations, simply known as Maxwell’s equations today, he was able to predict that waves of oscillating electric and magnetic fields travel in space at a particular speed, which he calculated was roughly equivalent to the speed of light (later, more accurate means of measurement confirmed exact equivalence). Subsequently, Maxwell theorized that light was just one of many possible types of electromagnetic radiation. Maxwell’s equations first appeared in 1864 in a paper entitled “A Dynamical Theory of the Electromagnetic Field,” but were more completely addressed in his Treatise on Electricity and Magnetism , published in 1873.

According to Maxwell’s theory (which, in its emphasis on fields, clearly opposed the theory of action at a distance that was popular at the time), electromagnetic waves should be able to be produced and studied in the laboratory. In fact, both infrared and ultraviolet radiation, which are just outside the visible electromagnetic spectrum, had already been discovered and investigated. However, it was not until William Heinrich Hertz discovered radio waves in 1887 that additional electromagnetic radiation even further outside the visible spectrum was proven to exist. In addition to anticipating this discovery, Maxwell’s theory greatly influenced the accepted understanding of the physical world and helped lead to Albert Einstein’s special theory of relativity and Max Planck’s quantum theory.

Maxwell resigned from King’s College in 1865 and relocated to the home in Scotland that been left to him by his father. He remained active in London academic circles, however, returning to England at least once every spring and continuing involvement in Cambridge University’s mathematical exams. Moreover, Maxwell continued his scientific work at home, completing much of his Treatise and penning works on gases, topology and heat theory during this time. He would become much more closely associated with Cambridge in 1871, when he was appointed the first Cavendish Professor of Physics at the institution. Included among his responsibilities in the new position were overseeing the foundation of the Cavendish Laboratory and editing the research papers of Henry Cavendish. Maxwell worked at the Cavendish Laboratory until 1879, when a bout with abdominal cancer, the same disease that had precipitated his mother’s death, rendered him too ill to continue. He died on November 5 of that year and was laid to rest in his native Scotland.

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Secrets of sunspots and solar magnetic fields investigated in nasa supercomputing simulations.

Frank Tavares

The Sun is much more than just a source of light for Earth – it’s a dynamic and complex star, with storms, flares, and movement causing it to change constantly. Magnetic fields govern most of the solar activity we can observe but how they do this is still poorly understood. New results based on simulations out of NASA’s Advanced Supercomputing facility at NASA’s Ames Research Center in California’s Silicon Valley are painting a more complete picture of one of the most prominent magnetically-driven solar features – a cycle of sunspot formation known as a “torsional oscillation.”

A computational analysis of data about the Sun’s structure and dynamics from two NASA spacecraft has revealed the strength of these torsional oscillations driven by the magnetic fields in the deep interior of the Sun are continuing to decline. This indicates that the current sunspot cycle may be weaker than the previous one, and the long-term trend of declining magnetic fields of the Sun is likely to continue. Such changes in the Sun’s interior may have impacts on space weather and the Earth’s atmosphere and climate.

The sunspot cycle begins when a sunspot begins to form at about 30 degrees latitude on the Sun’s surface. The formation zone then begins to migrate towards the equator. At its peak intensity, the Sun’s global magnetic field has its polar regions reversed – as if there were a positive and negative end of a magnet at each of the Sun’s poles, and they were switched. These 22-year variations are caused by dynamo processes inside the Sun. A dynamo process is when rotating, convecting, and electrically conducting fluid or plasma helps maintain a magnetic field. These deep magnetic fields are hidden, and can’t be observed directly, but their effects can be seen in the variations of solar rotation, creating a cyclical pattern of migrating flows across zones – the torsional oscillations. In some areas, this rotation speeds up or slows down, while in others it remains steady.

This analysis used data from two NASA missions, the Solar and Heliospheric Observatory and the Solar Dynamics Observatory. The Joint Science Operations Center at Stanford University processed data from 22 years of observations from both missions – more than five petabytes in total. NASA’s supercomputing facilities handled flow analysis, numerical modeling, and visualization that gave scientists a better look at this complex pattern.

Going forward, improvements to the data’s resolution, data analysis techniques, and simulation models will help merge models of the Sun’s magnetic fields with those of sunspot activity, advancing the understanding of how these processes impact the Sun’s deep interior. What happens with the Sun, including the processes beneath its surface, affects the space weather that impacts the entire solar system, including Earth. The more we know about the star that lights our home, the better we can understand its impacts on our home planet.

For news media:

Members of the news media interested in covering this topic should reach out to the  NASA Ames newsroom .

Author: Frank Tavares, NASA’s Ames Research Center

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The huge solar storm is keeping power grid and satellite operators on edge

Geoff Brumfiel

Willem Marx

NASA's Solar Dynamics Observatory captured this image of solar flares early Saturday afternoon. The National Oceanic and Atmospheric Administration says there have been measurable effects and impacts from the geomagnetic storm. Solar Dynamics Observatory hide caption

NASA's Solar Dynamics Observatory captured this image of solar flares early Saturday afternoon. The National Oceanic and Atmospheric Administration says there have been measurable effects and impacts from the geomagnetic storm.

Planet Earth is getting rocked by the biggest solar storm in decades – and the potential effects have those people in charge of power grids, communications systems and satellites on edge.

The National Oceanic and Atmospheric Administration says there have been measurable effects and impacts from the geomagnetic storm that has been visible as aurora across vast swathes of the Northern Hemisphere. So far though, NOAA has seen no reports of major damage.

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There has been some degradation and loss to communication systems that rely on high-frequency radio waves, NOAA told NPR, as well as some preliminary indications of irregularities in power systems.

"Simply put, the power grid operators have been busy since yesterday working to keep proper, regulated current flowing without disruption," said Shawn Dahl, service coordinator for the Boulder, Co.-based Space Weather Prediction Center at NOAA.

NOAA Issues First Severe Geomagnetic Storm Watch Since 2005

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"Satellite operators are also busy monitoring spacecraft health due to the S1-S2 storm taking place along with the severe-extreme geomagnetic storm that continues even now," Dahl added, saying some GPS systems have struggled to lock locations and offered incorrect positions.

NOAA's GOES-16 satellite captured a flare erupting occurred around 2 p.m. EDT on May 9, 2024.

As NOAA had warned late Friday, the Earth has been experiencing a G5, or "Extreme," geomagnetic storm . It's the first G5 storm to hit the planet since 2003, when a similar event temporarily knocked out power in part of Sweden and damaged electrical transformers in South Africa.

The NOAA center predicted that this current storm could induce auroras visible as far south as Northern California and Alabama.

Extreme (G5) geomagnetic conditions have been observed! pic.twitter.com/qLsC8GbWus — NOAA Space Weather Prediction Center (@NWSSWPC) May 10, 2024

Around the world on social media, posters put up photos of bright auroras visible in Russia , Scandinavia , the United Kingdom and continental Europe . Some reported seeing the aurora as far south as Mallorca, Spain .

The source of the solar storm is a cluster of sunspots on the sun's surface that is 17 times the diameter of the Earth. The spots are filled with tangled magnetic fields that can act as slingshots, throwing huge quantities of charged particles towards our planet. These events, known as coronal mass ejections, become more common during the peak of the Sun's 11-year solar cycle.

A powerful solar storm is bringing northern lights to unusual places

Usually, they miss the Earth, but this time, NOAA says several have headed directly toward our planet, and the agency predicted that several waves of flares will continue to slam into the Earth over the next few days.

While the storm has proven to be large, predicting the effects from such incidents can be difficult, Dahl said.

Shocking problems

The most disruptive solar storm ever recorded came in 1859. Known as the "Carrington Event," it generated shimmering auroras that were visible as far south as Mexico and Hawaii. It also fried telegraph systems throughout Europe and North America.

Stronger activity on the sun could bring more displays of the northern lights in 2024

While this geomagnetic storm will not be as strong, the world has grown more reliant on electronics and electrical systems. Depending on the orientation of the storm's magnetic field, it could induce unexpected electrical currents in long-distance power lines — those currents could cause safety systems to flip, triggering temporary power outages in some areas.

my cat just experienced the aurora borealis, one of the world's most radiant natural phenomena... and she doesn't care pic.twitter.com/Ee74FpWHFm — PJ (@kickthepj) May 10, 2024

The storm is also likely to disrupt the ionosphere, a section of Earth's atmosphere filled with charged particles. Some long-distance radio transmissions use the ionosphere to "bounce" signals around the globe, and those signals will likely be disrupted. The particles may also refract and otherwise scramble signals from the global positioning system, according to Rob Steenburgh, a space scientist with NOAA. Those effects can linger for a few days after the storm.

Like Dahl, Steenburgh said it's unclear just how bad the disruptions will be. While we are more dependent than ever on GPS, there are also more satellites in orbit. Moreover, the anomalies from the storm are constantly shifting through the ionosphere like ripples in a pool. "Outages, with any luck, should not be prolonged," Steenburgh said.

What Causes The Northern Lights? Scientists Finally Know For Sure

The radiation from the storm could have other undesirable effects. At high altitudes, it could damage satellites, while at low altitudes, it's likely to increase atmospheric drag, causing some satellites to sink toward the Earth.

The changes to orbits wreak havoc, warns Tuija Pulkkinen, chair of the department of climate and space sciences at the University of Michigan. Since the last solar maximum, companies such as SpaceX have launched thousands of satellites into low Earth orbit. Those satellites will now see their orbits unexpectedly changed.

"There's a lot of companies that haven't seen these kind of space weather effects before," she says.

The International Space Station lies within Earth's magnetosphere, so its astronauts should be mostly protected, Steenburgh says.

In a statement, NASA said that astronauts would not take additional measures to protect themselves. "NASA completed a thorough analysis of recent space weather activity and determined it posed no risk to the crew aboard the International Space Station and no additional precautionary measures are needed," the agency said late Friday.

People visit St Mary's lighthouse in Whitley Bay to see the aurora borealis on Friday in Whitley Bay, England. Ian Forsyth/Getty Images hide caption

People visit St Mary's lighthouse in Whitley Bay to see the aurora borealis on Friday in Whitley Bay, England.

While this storm will undoubtedly keep satellite operators and utilities busy over the next few days, individuals don't really need to do much to get ready.

"As far as what the general public should be doing, hopefully they're not having to do anything," Dahl said. "Weather permitting, they may be visible again tonight." He advised that the largest problem could be a brief blackout, so keeping some flashlights and a radio handy might prove helpful.

I took these photos near Ranfurly in Central Otago, New Zealand. Anyone can use them please spread far and wide. :-) https://t.co/NUWpLiqY2S — Dr Andrew Dickson reform/ACC (@AndrewDickson13) May 10, 2024

And don't forget to go outside and look up, adds Steenburgh. This event's aurora is visible much further south than usual.

A faint aurora can be detected by a modern cell phone camera, he adds, so even if you can't see it with your eyes, try taking a photo of the sky.

The aurora "is really the gift from space weather," he says.

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23.2: Electromagnetic Waves and their Properties

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learning objectives

• Explain the meaning and importance of Maxwell’s equations

Maxwell’s Equations

Maxwell’s equations are a set of four partial differential equations that, along with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits.

Named after esteemed physicist James Clerk Maxwell, the equations describe the creation and propagation of electric and magnetic fields. Fundamentally, they describe how electric charges and currents create electric and magnetic fields, and how they affect each other.

Maxwell’s equations can be divided into two major subsets. The first two, Gauss’s law and Gauss’s law for magnetism, describe how fields emanate from charges and magnets respectively. The other two, Faraday’s law and Ampere’s law with Maxwell’s correction, describe how induced electric and magnetic fields circulate around their respective sources.

Each of Maxwell’s equations can be looked at from the “microscopic” perspective, which deals with total charge and total current, and the “macroscopic” set, which defines two new auxiliary fields that allow one to perform calculations without knowing microscopic data like atomic-level charges.

Gauss’s Law

Gauss’s law relates an electric field to the charge(s) that create(s) it. The field (E) points towards negative charges and away from positive charges, and from the microscopic perspective, is related to charge density (ρ) and vaccuum permittivity (ε 0 , or permittivity of free space) as:

\[\nabla \cdot \mathbf { E } = \dfrac { \rho } { \epsilon _ { 0 } }\]

Gauss’s Law basically says that a net amount of charge contained within a region of space will generate an electric field that emanates through the surface that surrounds that region.

Example of Gauss’s Law : A positive charge contained within a region of space creates an electric field that emanates from the surface of that region.

Gauss’s Law for Magnetism

Gauss’s law for magnetism states that there are no “magnetic charges (or monopoles)” analogous to electric charges, and that magnetic fields are instead generated by magnetic dipoles . Such dipoles can be represented as loops of current, but in many ways are similar in appearance to positive and negative “magnetic charges” that are inseparable and thus have no formal net “magnetic charge.”

Magnetic field lines form loops such that all field lines that go into an object leave it at some point. Thus, the total magnetic flux through a surface surrounding a magnetic dipole is always zero.

Field lines caused by a magnetic dipole : The field lines created by this magnetic dipole either form loops or extend infinitely.

The differential form of Gauss’s law for magnetic for magnetism is

\[\nabla \cdot \mathbf { B } = \mathbf { 0 }\]

Faraday’s Law

Faraday’s law describes how a time-varying magnetic field (or flux) induces an electric field. The principle behind this phenomenon is used in many electric generators. Both macroscopic and microscopic differential equations are the same, relating electric field (E) to the time-partial derivative of magnetic field (B):

\[\nabla \times \mathbf { E } = - \frac { \partial \mathbf { B } } { \partial \mathbf { t } }\]

Ampere’s Circuital Law (with Maxwell’s correction)

Ampere’s law originally stated that magnetic field could be created by electrical current. Maxwell added a second source of magnetic fields in his correction: a changing electric field (or flux), which would induce a magnetic field even in the absence of an electrical current. He named the changing electric field “displacement current.”

Maxwell’s correction shows that self-sustaining electromagnetic waves (light) can travel through empty space even in the absence of moving charges or currents, with the electric field component and magnetic field component each continually changing and each perpetuating the other.

Electromagnetic Waves : Electric (red) and magnetic (blue) waves propagate in phase sinusoidally, and perpendicularly to one another.

The microscopic approach to the Maxwell-corrected Ampere’s law relates magnetic field (B) to current density (J, or current per unit cross sectional area) and the time-partial derivative of electric field (E):

\[\nabla \times \mathbf { B } = \mu _ { 0 } \mathbf { J } + \mu _ { 0 } \epsilon _ { 0 } \frac { \partial \mathbf { E } } { \partial t }\]

The Production of Electromagnetic Waves

Electromagnetic waves are the combination of electric and magnetic field waves produced by moving charges.

• Explain the self-perpetuating behavior of an electromagnetic wave

Electromagnetic waves

Electromagnetic radiation, is a form of energy emitted by moving charged particles. As it travels through space it behaves like a wave, and has an oscillating electric field component and an oscillating magnetic field. These waves oscillate perpendicularly to and in phase with one another.

Electromagnetic Wave : Electromagnetic waves are a self-propagating transverse wave of oscillating electric and magnetic fields. The direction of the electric field is indicated in blue, the magnetic field in red, and the wave propagates in the positive x-direction. Notice that the electric and magnetic field waves are in phase.

The creation of all electromagnetic waves begins with a charged particle. This charged particle creates an electric field (which can exert a force on other nearby charged particles). When it accelerates as part of an oscillatory motion, the charged particle creates ripples, or oscillations, in its electric field, and also produces a magnetic field (as predicted by Maxwell’s equations).

Once in motion, the electric and magnetic fields created by a charged particle are self-perpetuating—time-dependent changes in one field (electric or magnetic) produce the other. This means that an electric field that oscillates as a function of time will produce a magnetic field, and a magnetic field that changes as a function of time will produce an electric field. Both electric and magnetic fields in an electromagnetic wave will fluctuate in time, one causing the other to change.

Electromagnetic waves are ubiquitous in nature (i.e., light) and used in modern technology—AM and FM radio, cordless and cellular phones, garage door openers, wireless networks, radar, microwave ovens, etc. These and many more such devices use electromagnetic waves to transmit data and signals.

All the above sources of electromagnetic waves use the simple principle of moving charge, which can be easily modeled. Placing a coin in contact with both terminals of a 9-volt battery produces electromagnetic waves that can be detected by bringing the antenna of a radio (tuned to a static-producing station) within a few inches of the point of contact.

Energy and Momentum

Electromagnetic waves have energy and momentum that are both associated with their wavelength and frequency.

• Relate energy of an electromagnetic wave with the frequency and wavelength

Electromagnetic radiation can essentially be described as photon streams. These photons are strictly defined as massless, but have both energy and surprisingly, given their lack of mass, momentum, which can be calculated from their wave properties.

Waves were poorly understood until the 1900s, when Max Planck and Albert Einstein developed modern corrections to classical theory.

Planck theorized that “black bodies” (thermal radiators) and other forms of electromagnetic radiation existed not as spectra, but in discrete, “quantized” form. In other words, there were only certain energies an electromagnetic wave could have. In his work he developed what is now known as “Planck’s constant,” which is approximately equal to 6.626×10 -34 J·s.

The energy (E) of a photon can be related to its frequency (f) by Planck’s constant (h):

\[\mathrm { E } = \mathrm { hf } = \frac { \mathrm { hc } } { \lambda }\]

The ratio of speed of light (c) to wavelength (λ) can be substituted in place of f to give the same equation to energy in different terms. Note that energy cannot take any value: it can only exist in increments of frequency times Planck’s constant (or Planck’s constant times c divided by wavelength). Energy of a wave is therefore “quantized. ”

Wavelength : Wavelength of the sinusoidal function is represented by λ.

Momentum is classically defined as the product of mass and velocity and thus would intuitively seem irrelevant to a discussion of electromagnetic radiation, which is both massless and composed of waves.

However, Einstein proved that light can act as particles in some circumstances, and that a wave-particle duality exists. And, given that he related energy and mass (E=mc 2 ), it becomes more conceivable that a wave (which has an energy value) not only has an equation to mass but a momentum as well.

And indeed, Einstein proved that the momentum (p) of a photon is the ratio of its energy to the speed of light.

\[\mathrm { p } = \dfrac { \mathrm { E } } { \mathrm { c } } = \dfrac { \mathrm { hf } } { \mathrm { c } } = \dfrac { \mathrm { h } } { \lambda }\]

Substituting E with hc/λ cancels the c terms, making momentum also equal to the simple ratio of Planck’s constant to wavelength.

The Speed of Light

The speed of light in a vacuum is one of the most fundamental constant in physics, playing a pivotal role in modern physics.

• Relate speed of light with the index of refraction of the medium

The speed of light is generally a point of comparison to express that something is fast. shows a scale representation of the time it takes a beam of light to reach the moon from Earth. But what exactly is the speed of light?

Light Going from Earth to the Moon : A beam of light is depicted travelling between the Earth and the Moon in the time it takes a light pulse to move between them: 1.255 seconds at their mean orbital (surface-to-surface) distance. The relative sizes and separation of the Earth–Moon system are shown to scale.

It is just that: the speed of a photon or light particle. The speed of light in a vacuum (commonly written as c) is 299,792,458 meters per second. This is a universal physical constant used in many areas of physics. For example, you might be familiar with the equation:

\[\mathrm { E } = \mathrm { mc } ^ { 2 }\]

where E = Energy and m = mass. This is known as the mass-energy equivalence, and it uses the speed of light to interrelate space and time. This not only explains the energy a body of mass contains, but also explains the hindrance mass has on speed.

There are many uses for the speed of light in a vacuum, such as in special relativity, which says that c is the natural speed limit and nothing can move faster than it. However, we know from our understanding of physics (and previous atoms) that the speed at which something travels also depends on the medium through which it is traveling. The speed at which light propagates through transparent materials (air, glass, etc.,) is dependent on the refractive index of that material, n:

\[\mathrm { v } = \dfrac { \mathrm { c } } { \mathrm { n } }\]

where v = actual velocity of light moving through the medium, c = speed of light in a vacuum, and n = refractive index of medium. The refractive index of air is about 1.0003, and from this equation we can find that the speed of visible light in air is about 90 km/s slower than c.

As mentioned earlier, the speed of light (usually of light in a vacuum) is used in many areas of physics. Below is an example of an application of the constant c.

The Lorentz Factor

Fast-moving objects exhibit some properties that are counterintuitive from the perspective of classical mechanics. For example, length contracts and time dilates (runs slower) for objects in motion. The effects are typically minute, but are noticeable at sufficiently high speeds. The Lorentz factor (γ) is the factor by which length shortens and time dilates as a function of velocity (v):

\[\gamma = \left( 1 - \mathrm{ v } ^ { 2 } / \mathrm { c } ^ { 2 } \right) ^ { - 1 / 2 } \gamma = \left( 1 - \mathrm { v } ^ { 2 } / \mathrm { c } ^ { 2 } \right) ^ { - 1 / 2 } \gamma = \left( 1 - \mathrm { v } ^ { 2 } / \mathrm { c } ^ { 2 } \right) ^ { - 1 / 2 }\]

At low velocities, the quotient of v 2 /c 2 is sufficiently close to 0 such that γ is approximately 1. However, as velocity approaches c, γ increases rapidly towards infinity.

The Doppler Effect

The Doppler Effect is the change in a wave’s perceived frequency that results from the source’s motion, the observer, and the medium.

• Give examples of daily observations of the Doppler effect

The Doppler effect is a periodic event’s change in frequency for an observer in motion relative to the event’s source. Typically, this periodic event is a wave.

Most people have experienced the Doppler effect in action. Consider an emergency vehicle in motion, sounding its siren. As it approaches an observer, the pitch of the sound (its frequency) sounds higher than it actually is. When the vehicle reaches the observer, the pitch is perceived as it actually is. When the vehicle continues away from the observer, the pitch is perceived as lower than it actually is. From the perspective of an observer inside the vehicle, the pitch of the siren is constant.

The Doppler Effect and Sirens : Waves emitted by a siren in a moving vehicle

The difference in the perceived pitch depending on observer location can be explained by the fact that the siren’s position changes as it emits waves. A wave of sound is emitted by a moving vehicle every millisecond. The vehicle ‘chases’ each wave in one direction. By the time the next wave is emitted, it is closer (relative to an onlooker ahead of the vehicle) to the previous wave than the wave’s frequency would suggest. Relative to an onlooker behind the vehicle, the second wave is further from the first wave than one would expect, which suggests a lower frequency.

The Doppler effect can be caused by any kind of motion. In the example above, the siren moved relative to a stationary observer. If the observer moves relative to the stationary siren, the observer will notice the Doppler effect on the pitch of the siren. Finally, if the medium through which the waves propagate moves, the Doppler effect will be noticed even for a stationary observer. An example of this phenomenon is wind.

Quantitatively, the Doppler effect can be characterized by relating the frequency perceived (f) to the velocity of waves in the medium (c), the velocity of the receiver relative to the medium (v r ), the velocity of the source relative to the medium (v s ), and the actual emitted frequency (f 0 ):

\[\mathrm { f } = \left( \dfrac { \mathrm { c } + \mathrm { v } _ { \mathrm { r } } } { \mathrm { c } + \mathrm { v } _ { \mathrm { s } } } \right) \mathrm { f } _ { 0 }\]

The Doppler Effect : Wavelength change due to the motion of source

Momentum Transfer and Radiation Pressure Atom

Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic (EM) radiation.

• Explain formation of radiation pressure

Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic (EM) radiation. EM radiation (or photon, which is a quantum of light) carries momentum; this momentum is transferred to an object when the radiation is absorbed or reflected. Perhaps one of the most well know examples of the radiation pressure would be comet tails. Haley’s comet is shown in.

Halley’s Comet : As a comet approaches the inner Solar System, solar radiation causes the volatile materials within the comet to vaporize and stream out of the nucleus. The streams of dust and gas thus released form an atmosphere around the comet (called the coma), and the force exerted on the coma by the Sun’s radiation pressure and solar wind cause the formation of an enormous tail that points away from the Sun.

Although radiation pressure can be understood using classical electrodynamics, here we will examine the quantum mechanical argument. From the perspective of quantum theory, light is made of photons: particles with zero mass but which carry energy and – importantly in this argument – momentum. According to special relativity, because photons are devoid of mass, their energy (E) and momentum (p) are related by E=pc.

Now consider a beam of light perpendicularly incident on a surface, and let us assume the beam of light is totally absorbed. The momentum the photons carry is a conserved quantity (i.e., it cannot be destroyed) so it must be transferred to the surface; thus the absorption of the light beam causes the surface to gain momentum. Newton’s Second Law tells us that force equals rate of change of momentum; thus during each second, the surface experiences a force (or pressure, as pressure is force per unit area) due to the momentum the photons transfer to it.

This gives us: pressure = momentum transferred per second per unit area = energy deposited per second per unit area / c = I/c, (where I is the intensity of the beam of light).

Laser Cooling

There are many variations of laser cooling, but they all use radiation pressure to remove energy from atomic gases (and therefore cool the sample). In laser cooling (sometimes called Doppler cooling), the frequency of light is tuned slightly below an electronic transition in the atom. Because light is detuned to the “red” (i.e., at lower frequency) of the transition, the atoms will absorb more photons if they move towards the light source, due to the Doppler effect. Thus if one applies light from two opposite directions, the atoms will always scatter more photons from the laser beam pointing opposite to their direction of motion (typical setups applies three opposing pairs of laser beams as in ).

The Magneto Optical Trap : Experimental setup of Magneto Optical Trap (MOT), which uses radiation pressure to cool atomic species. Atoms are slowed down by absorbing (and emitting) photons.

In each scattering event, the atom loses a momentum equal to the momentum of the photon. If the atom (which is now in the excited state) then emits a photon spontaneously, it will be kicked by the same amount of momentum, only in a random direction. Since the initial momentum loss was opposite to the direction of motion (while the subsequent momentum gain was in a random direction), the overall result of the absorption and emission process is to reduce the speed of the atom. If the absorption and emission are repeated many times, the average speed (and therefore the kinetic energy ) of the atom will be reduced. Since the temperature of a group of atoms is a measure of the average random internal kinetic energy, this is equivalent to cooling the atoms. Simple laser cooling setups can produce a cold sample of atomic gases at around 1mK (=10 -3 K) starting from a room temperature gas.

• Maxwell’s four equations describe how electric charges and currents create electric and magnetic fields, and how they affect each other.
• Gauss’s law relates an electric field to the charge(s) that create(s) it.
• Gauss’s law for magnetism states that there are no “magnetic charges” analogous to electric charges, and that magnetic fields are instead generated by magnetic dipoles.
• Faraday’s law describes how a time-varying magnetic field (or flux ) induces an electric field. The principle behind this phenomenon is used in many electric generators.
• Ampere ‘s law originally stated that a magnetic field is created by an electrical current. Maxwell added that a changing electric flux can also generate a magnetic field.
• Electromagnetic waves consist of both electric and magnetic field waves. These waves oscillate in perpendicular planes with respect to each other, and are in phase.
• The creation of all electromagnetic waves begins with an oscillating charged particle, which creates oscillating electric and magnetic fields.
• Once in motion, the electric and magnetic fields that a charged particle creates are self-perpetuating: time-dependent changes in one field (electric or magnetic) produce the other.
• Max Planck proved that energy of a photon (a stream of which is an electromagnetic wave ) is quantized and can exist in multiples of “Planck’s constant” (denoted as h, approximately equal to 6.626×10 -34 J·s).
• \(\mathrm { E } = \mathrm { hf } = \frac { \mathrm { hc } } { \lambda } \)describes the energy (E) of a photon as a function of frequency (f), or wavelength (λ).
• \(\mathrm { p } = \frac { \mathrm { E } } { \mathrm { c } } = \frac { \mathrm { hf } } { \mathrm { c } } = \frac { \mathrm { h } } { \lambda }\) describes the momentum (p) of a photon as a function of its energy, frequency, or wavelength.
• The maximum possible value for the speed of light is that of light in a vacuum, and this speed is used for a constant in many area of physics.
• c is the symbol used to represent the speed of light in a vacuum, and its value is 299,792,458 meters per second.
• When light travels through medium, its speed is hindered by the index of refraction of that medium. Its actual speed can be found with: \(v=\frac{c}{n}\).
• The Doppler effect is very commonly observed in action.
• The Doppler effect can be observed in the apparent change in pitch of a siren on an emergency vehicle, according to a stationary observer.
• The observer will notice the Doppler effect on the pitch of the stationary siren when moving relative to its pitch, or if the medium moves when the observer is stationary.
• Photons carry momentum (p = E/c). When photons are absorbed or reflected on a surface, the surface receives momentum kicks. This momentum transfer leads to radiation pressure.
• Electromagnetic radiation applies radiation pressure equal to the Intensity (of light beam) divided by c (speed of light).
• Laser cooling uses radiation pressure to remove energy from atomic gases. The technique can produce cold samples of gases at 1mK or so.
• differential equation : An equation involving the derivatives of a function.
• flux : A quantitative description of the transfer of a given vector quantity through a surface. In this context, we refer to the electric flux and magnetic flux.
• electromagnetic wave : A wave of oscillating electric and magnetic fields.
• phase : Waves are said to be “in phase” when they begin at the same part (e.g., crest) of their respective cycles.
• photon : The quantum of light and other electromagnetic energy, regarded as a discrete particle having zero rest mass, no electric charge, and an indefinitely long lifetime.
• wavelength : The length of a single cycle of a wave, as measured by the distance between one peak or trough of a wave and the next; it is often designated in physics as λ, and corresponds to the velocity of the wave divided by its frequency.
• frequency : The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.
• special relativity : A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.
• refractive index : The ratio of the speed of light in air or vacuum to that in another medium.
• doppler effect : Apparent change in frequency of a wave when the observer and the source of the wave move relative to each other.
• classical electrodynamics : A branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents.

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I’m a Comfy Shoe Snob, and These Are the 8 Travel Styles I Rotate Through My Suitcase — From $30

From my go-to adventure sandals to the only rain boots worth traveling with, these shoes literally keep me on my toes.

Travel + Leisure / Marcus Millan

I am one of those travelers who would rather walk the 2.5 miles from Saint-Germain-des-Prés to Montmartre than take the metro, just to soak up the feel of every arrondissement and sample a chocolate croissant from any Parisian patisserie I come by along the way. I am a firm believer in exploring on foot, and I’ve had enough trip-shattering blisters and cracks in my heels to know which shoes are cut out for the job and which aren’t.

As a tireless walker , sandal enthusiast, and style-driven comfy shoe snob (truly a restrictive combination), I’ve tried the gamut and narrowed my current rotation to these eight tried-and-true styles.

Chaco Z Sandals

The first time I wore these Chaco Z sandals was on an hour-long hike on varied terrain in the Arizona desert. They gripped the sandstone and provided a cushiony barrier between my feet and the rock. Hiking boots would have been covered in red sand, impossible to clean off, but I threw these Chacos in the sink of my hotel room and voila! They were clean and dry by dinner. I daresay they’re some of the comfiest adventure sandals I’ve ever worn.

Reebok Women's Club C Walking Shoes

A versatile and comfy plain-ish white sneaker is a travel must-have in my book. The Club C Reebok style with over 800 five-star ratings on Amazon is suitable for long city jaunts, but it looks a little cooler than your average athletic shoe. The retro aesthetic goes well with any vintage-inspired denim. My one word of warning: Just don’t get them wet. 

Blundstone Vegan Chelsea Boots

They’re on the pricey side compared to other non-leather Chelsea boots you might find on Amazon, but believe me — your feet will thank you for spending a little extra on quality materials that are soft, breathable, nonrestrictive, shock-absorbing, water-resistant, and everlasting. Mine are three years old now with hardly any signs of wear. I love how comfortable they are and that they move with my feet, even though they aren’t made of genuine leather.

Merrell Moab Speed 2 Hiking Shoes

I’m an avid hiker who’s cycled through several iterations of the now-classic Merrell Moab boot. One of the more recent versions, the Moab Speed 2, is my current go-to. It has all the beloved features of the original Moab but with 30 percent more foam in the midsole and Vibram TC5+ outsole technology for improved traction and debris-shedding. And the fun colors are a major plus.

Hunter Original Play Short Boots

Rain boots are a staple of my wardrobe at home, but I wouldn’t think to travel with them — I mean, considering the size of them … and the weight! — were it not for Hunter’s Play Short style. Hunter-brand wellingtons are known and loved for their durability. I bought my first pair more than 10 years ago and they’re as waterproof now as they were a decade ago, when I was a college student rushing between classes in the rain.

Teva Original Universal Sandal

While I love my Chacos, the Teva Original Universal Sandal was the first adventure sandal to steal my heart when I was a twentysomething backpacking in New Zealand. I wore them so frequently that my feet tanned around the distinctive velcro straps (“Teva tan”). I’m now on my third pair, and many Amazon reviewers agree with me that they’re some of the best sandals to travel in .

Cushionaire Cork Sandals

These Birkenstock lookalikes are super comfy, affordable, and made of vegan materials. They have more than 40,000 five-star ratings on Amazon, making them the No. 1 best-seller in women’s slides. At least one pair is sold every hour, on average. They’re ideal for slipping on and walking around in at the hotel.

Brooks Adrenaline GTS 22 Running Shoe

A versatile tennis shoe is good to always have in your suitcase. You can go for a morning run through the city, take a cool dance class, squeeze in a session at the hotel gym, or go for a hike in these. I have brought them with me on trips when I’ve been amid marathon training, and they’ve kept my feet strong enough to walk around comfortably even after miles of running.

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