Maxwell's Equations: Definition, Derivation, How to Remember (w/ Examples) (2023)

Solving the mysteries of electromagnetism has been one of the greatest accomplishments of physics to date, and the lessons learned are fully encapsulated in Maxwell’s equations.

James Clerk Maxwell gives his name to these four elegant equations, but they are the culmination of decades of work by many physicists, including Michael Faraday, Andre-Marie Ampere and Carl Friedrich Gauss – who give their names to three of the four equations – and many others. While Maxwell himself only added a term to one of the four equations, he had the foresight and understanding to collect the very best of the work that had been done on the topic and present them in a fashion still used by physicists today.

For many, many years, physicists believed electricity and magnetism were separate forces and distinct phenomena. But through the experimental work of people like Faraday, it became increasingly clear that they were actually two sides of the same phenomenon, and Maxwell’s equations present this unified picture that is still as valid today as it was in the 19th century. If you’re going to study physics at higher levels, you absolutely need to know Maxwell’s equations and how to use them.

Maxwell’s Equations

Maxwell’s equations are as follows, in both the differential form and the integral form. (Note that while knowledge of differential equations is helpful here, a conceptual understanding is possible even without it.)

Gauss’ Law for Electricity

Differential form:

\bm{∇∙E} = \frac{ρ}{ε_0}

Integral form:

\int \bm{E ∙} d\bm{A} = \frac{q}{ε_0}

No Monopole Law / Gauss’ Law for Magnetism

Differential form:

\bm{∇∙B} = 0

Integral form:

\int \bm{B ∙} d\bm{A} = 0

Faraday’s Law of Induction

(Video) Maxwell's Equations - Basic derivation

Differential form:

\bm{∇ × E} = − \frac{∂\bm{B}}{∂t}

Integral form:

\int \bm{E∙ }d\bm{s}= − \frac{∂\phi_B}{ ∂t}

Ampere-Maxwell Law / Ampere’s Law

Differential form:

\bm{∇ × B} = \frac{J}{ ε_0 c^2} + \frac{1}{c^2} \frac{∂E}{∂t}

Integral form:

\int \bm{B ∙} d\bm{s} = μ_0 I + \frac{1}{c^2} \frac{∂}{∂t} \int \bm{E ∙ }d\bm{A}

Symbols Used in Maxwell’s Equations

Maxwell’s equations use a pretty big selection of symbols, and it’s important you understand what these mean if you’re going to learn to apply them. So here’s a run-down of the meanings of the symbols used:

B​ = magnetic field

E​ = electric field

ρ​ = electric charge density

ε0​ = permittivity of free space = 8.854 × 10-12 m-3 kg-1 s4 A2

q​ = total electric charge (net sum of positive charges and negative charges)

(Video) Maxwell Equations in differential and integral form| Basic laws to understand EMFT Maxwell Equations

𝜙B = magnetic flux

J​ = current density

I​ = electric current

c​ = speed of light = 2.998 × 108 m/s

μ0 = permeability of free space = 4π × 10−7 N / A2

Additionally, it’s important to know that ∇ is the del operator, a dot between two quantities (​X​ ∙ ​Y​) shows a scalar product, a bolded multiplication symbol between two quantities is a vector product (​X​ × ​Y​), that the del operator with a dot is called the “divergence” (e.g., ∇ ∙​X​ = divergence of ​X​ = div ​X​) and a del operator with a scalar product is called the curl (e.g., ∇ ​×​ ​Y​ = curl of ​Y​ = curl ​Y​). Finally, the ​A​ in d​A​ means the surface area of the closed surface you’re calculating for (sometimes written as d​S​), and the ​s​ in d​s​ is a very small part of the boundary of the open surface you’re calculating for (although this is sometimes d​l​, referring to an infinitesimally small line component).

Derivation of the Equations

The first equation of Maxwell’s equations is Gauss’ law, and it states that the net electric flux through a closed surface is equal to the total charge contained inside the shape divided by the permittivity of free space. This law can be derived from Coulomb’s law, after taking the important step of expressing Coulomb’s law in terms of an electric field and the effect it would have on a test charge.

The second of Maxwell’s equations is essentially equivalent to the statement that “there are no magnetic monopoles.” It states that the net magnetic flux through a closed surface will always be 0, because magnetic fields are always the result of a dipole. The law can be derived from the Biot-Savart law, which describes the magnetic field produced by a current element.

The third equation – Faraday’s law of induction – describes how a changing magnetic field produces a voltage in a loop of wire or conductor. It was originally derived from an experiment. However, given the result that a changing magnetic flux induces an electromotive force (EMF or voltage) and thereby an electric current in a loop of wire, and the fact that EMF is defined as the line integral of the electric field around the circuit, the law is easy to put together.

The fourth and final equation, Ampere’s law (or the Ampere-Maxwell law to give him credit for his contribution) describes how a magnetic field is generated by a moving charge or a changing electric field. The law is the result of experiment (and so – like all of Maxwell’s equations – wasn’t really “derived” in a traditional sense), but using ​Stokes’ theorem​ is an important step in getting the basic result into the form used today.

Examples of Maxwell’s Equations: Gauss’ Law

To be frank, especially if you aren’t exactly up on your vector calculus, Maxwell’s equations look quite daunting despite how relatively compact they all are. The best way to really understand them is to go through some examples of using them in practice, and Gauss’ law is the best place to start. Gauss’ law is essentially a more fundamental equation that does the job of Coulomb’s law, and it’s pretty easy to derive Coulomb’s law from it by considering the electric field produced by a point charge.

Calling the charge ​q​, the key point to applying Gauss’ law is choosing the right “surface” to examine the electric flux through. In this case, a sphere works well, which has surface area ​A​ = 4π​r2, because you can center the sphere on the point charge. This is a huge benefit to solving problems like this because then you don’t need to integrate a varying field across the surface; the field will be symmetric around the point charge, and so it will be constant across the surface of the sphere. So the integral form:

\int \bm{E ∙} d\bm{A} = \frac{q}{ε_0}

Can be expressed as:

E × 4πr^2 = \frac{q}{ε_0}

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Note that the ​E​ for the electric field has been replaced with a simple magnitude, because the field from a point charge will simply spread out equally in all directions from the source. Now, dividing through by the surface area of the sphere gives:

E = \frac{q}{4πε_0r^2}

Since the force is related to the electric field by ​E​ = ​F​/​q​, where ​q​ is a test charge, ​F​ = ​qE​, and so:

F = \frac{q_1q_2}{4πε_0r^2}

Where the subscripts have been added to differentiate the two charges. This is Coulomb’s law stated in standard form, shown to be a simple consequence of Gauss’ law.

Examples of Maxwell’s Equations: Faraday’s Law

Faraday’s law allows you to calculate the electromotive force in a loop of wire resulting from a changing magnetic field. A simple example is a loop of wire, with radius ​r​ = 20 cm, in a magnetic field that increases in magnitude from ​Bi = 1 T to ​Bf = 10 T in the space of ∆​t​ = 5 s – what is the induced EMF in this case? The integral form of the law involves the flux:

\int \bm{E∙ }d\bm{s}= − \frac{∂\phi_B}{ ∂t}

which is defined as:

ϕ = BA \cos (θ)

The key part of the problem here is finding the rate of change of flux, but since the problem is fairly straightforward, you can replace the partial derivative with a simple “change in” each quantity. And the integral really just means the electromotive force, so you can rewrite Faraday’s law of induction as:

\text{EMF} = − \frac{∆BA \cos (θ)}{∆t}

If we assume the loop of wire has its normal aligned with the magnetic field, ​θ​ = 0° and so cos (​θ​) = 1. This leaves:

\text{EMF} = − \frac{∆BA}{∆t}

The problem can then be solved by finding the difference between the initial and final magnetic field and the area of the loop, as follows:

\begin{aligned} \text{EMF} &= − \frac{∆BA}{∆t} \\ &= − \frac{(B_f - B_i) × πr^2}{∆t} \\ &= − \frac{(10 \text{ T}- 1 \text{ T}) × π × (0.2 \text{ m})^2}{5 \text{ s}} \\ &= − 0.23 \text{ V} \end{aligned}

(Video) Maxwell's Equation - derivation - thermodynamics

This is only a small voltage, but Faraday’s law is applied in the same way regardless.

Examples of Maxwell’s Equations: Ampere-Maxwell Law

The Ampere-Maxwell law is the final one of Maxwell’s equations that you’ll need to apply on a regular basis. The equation reverts to Ampere’s law in the absence of a changing electric field, so this is the easiest example to consider. You can use it to derive the equation for a magnetic field resulting from a straight wire carrying a current ​I​, and this basic example is enough to show how the equation is used. The full law is:

\int \bm{B ∙} d\bm{s} = μ_0 I + \frac{1}{c^2} \frac{∂}{∂t} \int \bm{E ∙ }d\bm{A}

But with no changing electric field it reduces to:

\int \bm{B ∙} d\bm{s} = μ_0 I

Now, as with Gauss’ law, if you choose a circle for the surface, centered on the loop of wire, intuition suggests that the resulting magnetic field will be symmetric, and so you can replace the integral with a simple product of the circumference of the loop and the magnetic field strength, leaving:

B × 2πr = μ_0 I

Dividing through by 2π​r​ gives:

B = \frac{μ_0 I}{2πr}

Which is the accepted expression for the magnetic field at a distance ​r​ resulting from a straight wire carrying a current.

Electromagnetic Waves

When Maxwell assembled his set of equations, he began finding solutions to them to help explain various phenomena in the real world, and the insight it gave into light is one of the most important results he obtained.

Because a changing electric field generates a magnetic field (by Ampere’s law) and a changing magnetic field generates an electric field (by Faraday’s law), Maxwell worked out that a self-propagating electromagnetic wave might be possible. He used his equations to find the wave equation that would describe such a wave and determined that it would travel at the speed of light. This was a “eureka” moment of sorts; he realized that light is a form of electromagnetic radiation, working just like the field he imagined!

An electromagnetic wave consists of an electric field wave and a magnetic field wave oscillating back and forth, aligned at right angles to each other. The oscillation of the electric part of the wave generates the magnetic field, and the oscillating of this part in turn produces an electric field again, on and on as it travels through space.

Like any other wave, an electromagnetic wave has a frequency and a wavelength, and the product of these is always equal to ​c​, the speed of light. Electromagnetic waves are all around us, and as well as visible light, other wavelengths are commonly called radio waves, microwaves, infrared, ultraviolet, X-rays and gamma rays. All of these forms of electromagnetic radiation have the same basic form as explained by Maxwell’s equations, but their energies vary with frequency (i.e., a higher frequency means a higher energy).

So, for a physicist, it was Maxwell who said, “Let there be light!”

(Video) Physics - E&M: Maxwell's Equations (1 of 30) What are the Maxwell equations? Introduction


How do you memorize Maxwell's equations? ›

A mnemonic used by students to remember the Maxwell relations (in thermodynamics) is "Good Physicists Have Studied Under Very Fine Teachers", which helps them remember the order of the variables in the square, in clockwise direction.

What is Maxwells equation and its derivation? ›

curl H = dD/dt + J.

Maxwell's equation illustrated the speed of electromagnetic waves is the same as the speed of light. This is used in understanding the principle of antennas. The flow of electric current produces a magnetic field. When the flow of charges varies with time, it induces an electric field.

What are the 4 Maxwell's equations What do they mean and what are their mathematical equation equivalent? ›

The four Maxwell equations, corresponding to the four statements above, are: (1) div D = ρ, (2) div B = 0, (3) curl E = -dB/dt, and (4) curl H = dD/dt + J.

How do you explain Maxwell's equations? ›

Maxwell's equations are a set of four equations that describe the behavior of electric and magnetic fields and how they relate to each other. Ultimately they demonstrate that electric and magnetic fields are two manifestations of the same phenomenon. where c is recognized as the speed of light.

What is the easiest way to memorize equations? ›

8 Easy Ways to Memorize Math Formulas
  1. Develop a Keen Interest in the Subject.
  2. Employ Visual Memory.
  3. Try Different Learning Strategies.
  4. Get Rid of Distractions.
  5. Practice as Much as You can.
  6. Use Memory Techniques.
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  8. Understand the Formulas.
Jul 20, 2022

What is the easiest way to memorize formulas? ›

8 ways to memorise math formulas in easy way
  1. advertisement.
  2. Grow your interest in the concept which you are studying.
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Dec 22, 2021

Why are there exactly 4 Maxwell equations? ›

Maxwell's Equations are composed of four equations with each one describes one phenomenon respectively. Maxwell didn't invent all these equations, but rather he combined the four equations made by Gauss (also Coulomb), Faraday, and Ampere.

What are D and H in Maxwell's equations? ›


Here, D is the electric displacement, H is the magnetic vector, σ is the specific conductivity, ε is the dielectric constant (or permittivity), and μ is the magnetic permeability.

What is D and B in Maxwell equation? ›

The quantities D and B are the electric and magnetic flux densities and are in units of [coulomb/m2] and [weber/m2], or [tesla]. D is also called the electric displacement, and B, the magnetic induction.

What are the 4 laws of magnetism? ›

Electromagnetism: Faraday's law, Ampere's law, Lenz' law, & Lorentz force.

What are the symbols in Maxwell's equations? ›

Maxwell's Equations
Symbols Used
E = Electric fieldρ = charge densityi = electric current
B = Magnetic fieldε0 = permittivityJ = current density
D = Electric displacementμ0 = permeabilityc = speed of light
H = Magnetic field strengthM = MagnetizationP = Polarization

How many equations are there in Maxwell's equations? ›

Although there are just four today, Maxwell actually derived 20 equations in 1865. Later, Oliver Heaviside simplified them considerably. Using vector notation, he realised that 12 of the equations could be reduced to four – the four equations we see today.

Why is Maxwell equations so important? ›

Maxwell's equations are sort of a big deal in physics. They're how we can model an electromagnetic wave—also known as light. Oh, it's also how most electric generators work and even electric motors. Essentially, you are using Maxwell's equations right now, even if you don't know it.

What is first Maxwell equation? ›

∇⋅D=ρ. This is the first of Maxwell's equations.

What is the importance of Maxwell equation? ›

1 What is the main importance of Maxwell equations? Answer: Maxwell equations give us the idea that a changing magnetic field always induces an electric field and a changing electric field always induces a magnetic field.

What are the 3 rules to solving an equation? ›

Section 3: A General Rule for Solving Equations
  • Simplify each side of the equation by removing parentheses and combining like terms.
  • Use addition or subtraction to isolate the variable term on one side of the equation.
  • Use multiplication or division to solve for the variable.

What is the 4 basic rule in solving equation? ›

We have 4 ways of solving one-step equations: Adding, Substracting, multiplication and division. If we add the same number to both sides of an equation, both sides will remain equal. If we subtract the same number from both sides of an equation, both sides will remain equal.

What is the fastest way to memorize math facts? ›

Flashcards are a great no-fuss way to learn math facts. Shuffling a deck makes it easy to mix up the order in which students practice, a learner can study them together with a tutor, a peer, or on their own, and flashcards provide both visual stimuli and a chance for kinesthetic learning.

What is the fastest way to memorize in 10 minutes? ›

10 Ways to Memorizing Anything in Under 10 Minutes!
  1. Change up your environment. ...
  2. Associate words with visuals. ...
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  4. To-do List Applications. ...
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Sep 30, 2022

Why can't I remember maths? ›

What is dyscalculia? Dyscalculia is a learning disorder that affects a person's ability to understand number-based information and math. People who have dyscalculia struggle with numbers and math because their brains don't process math-related concepts like the brains of people without this disorder.

Why is Maxwell's theory so hard to understand? ›

Physicists found it hard to understand because the equations were complicated. Mathematicians found it hard to understand because Maxwell used physical language to explain it. It was regarded as an obscure speculation without much experimental evidence to support it.

Why Gauss law is called Maxwell's equation? ›

The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

Why is Maxwell equation linear? ›

The linearity of Maxwell's equations accounts for the well-known fact that the electric fields generated by point charges, as well as the magnetic fields generated by line currents, are superposable. , is clearly a statement of the conservation of electric charge.

Are all four Maxwell's equations independent? ›

It is now widely accepted that the Maxwell equations of Electrodynamics constitute a self-consistent set of four independent partial differential equations.

What are the four thermodynamic relations of Maxwell? ›

The thermodynamic parameters are: T (temperature), S (entropy), P (pressure), and V (volume).

What are the four Maxwell equations and Lorentz force law? ›

The four Maxwell's equations and the Lorentz force law (which together constitution the fundations of all the classical electromagnetism) are listed below: <br> (i) `oint vecB. vec(ds)=q//(in_0)` <br> (ii) `oint vecB. vec(ds)=0` <br> (iii) `oint vecE.

What is Maxwell's equation for light? ›

c = 1/(e0m0)1/2 = 2.998 X 108m/s. Light is an electromagnetic wave: this was realized by Maxwell circa 1864, as soon as the equation c = 1/(e0m0)1/2 = 2.998 X 108m/s was discovered, since the speed of light had been accurately measured by then, and its agreement with c was not likely to be a coincidence.

What is D in flux? ›

(the electric field, E, multiplied by the component of area perpendicular to the field). The electric flux over a surface S is therefore given by the surface integral: where E is the electric field and dS is a differential area on the closed surface S with an outward facing surface normal defining its direction.

What does D formula mean? ›

"d" stands for derivative.

What are the 3 main magnetic elements? ›

Since then only three elements on the periodic table have been found to be ferromagnetic at room temperature—iron (Fe), cobalt (Co), and nickel (Ni).

What are the 3 types of magnetic force? ›

The three types of magnets are temporary, permanent, and electromagnets.

What are the variables in Maxwell's equations? ›

Here, E is the total electric field, B is the total magnetic field, ρ is the electric charge density, J is the electric current density, ϵ0 is the permittivity in free space, μ0 is the permeability of free space, ∇⋅() is the divergence operator, and ∇×() is the curl operator.

What are the units of Maxwell equations? ›

They are measured in units of [coulomb/m3] and [ampere/m2].

What does the triangle mean in Maxwell's equations? ›

The triangle and dot symbol in front of the field symbols (called the "divergence" operator) is a mathematical way to measure if a field behaves as a source or a sink at a specific point in space.

Who simplified Maxwell's equations? ›

Heaviside championed the Faraday-Maxwell approach to electromagnetism and simplified Maxwell's original set of 20 equations to the four used today. Importantly, Heaviside rewrote Maxwell's Equations in a form that involved only electric and magnetic fields.

Are Maxwell's equations always true? ›

Conclusion: Maxwell equations are only true with some probability, depending whether or not a male wave can marry with a female wave. That is the reason in QET, the wave is probability wave.

What is 2nd Maxwell equation? ›

Therefore the net flux out of the enclosed volume is zero, Maxwell's second equation: ∫→B⋅d→A=0. The first two Maxwell's equations, given above, are for integrals of the electric and magnetic fields over closed surfaces .

What is the second equation of Maxwell? ›

Maxwell's second equation is the differential form of Gauss's law of magnetism. As magnetic, monopoles do not exist in magnets and the magnetic field lines form closed loops. There is no source of the magnetic field from which the lines will either only diverge or only converge.

How many Maxwell's are there? ›

In 1881 the official census of Great Britain records that there are 7200 Maxwells. Today I estimate there to be about 16,500 Maxwell in Scotland England Wales and Northern Ireland.

What is the fastest way to memorize physics equations? ›

Here are some handy methods to effectively memorize your Physics formulas.
  1. Understand the concept and meaning of each formula. ...
  2. Apply and practice. ...
  3. Make notes and paste them up. ...
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  6. Take care of your body.

Did Einstein use Maxwell's equations? ›

Albert Einstein dismissed the notion of the aether as an unnecessary one, and he concluded that Maxwell's equations predicted the existence of a fixed speed of light, independent of the velocity of the observer. Hence, he used the Maxwell's equations as the starting point for his Special Theory of Relativity.

What did Einstein say about Maxwell? ›

'The work of James Clerk Maxwell changed the world forever', said Albert Einstein. Einstein's comments referred to Maxwell's four great electromagnetics papers, published when he was Professor of Natural Philosophy at King's (1861-5).

How can I memorize 10x fast? ›

6 Tips on How to Memorize Fast and Easily
  1. Understand your learning style.
  2. Learn the 3 'R's of memorization.
  3. Practice the substitution method.
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Feb 12, 2021

How can I memorize 10 minutes? ›

10 Ways to Memorizing Anything in Under 10 Minutes!
  1. Change up your environment. ...
  2. Associate words with visuals. ...
  3. Memorize with the help of repetition. ...
  4. To-do List Applications. ...
  5. Use mnemonic devices. ...
  6. Make use of your senses. ...
  7. Don't wait until the last minute. ...
  8. Use time recording software.
Sep 30, 2022

What is the easiest way to answer physics questions? ›

  1. Focus on the Problem. Establish a clear mental image of the problem. A. ...
  2. Describe the Physics. Refine and quantify your mental image of the problem. A. ...
  3. Plan a Solution. Turn the concepts into math. A. ...
  4. Execute the Plan. This is the easiest step – it's just the algebra/calculus/etc. A. ...
  5. Evaluate the Answer. Be skeptical.

Why are Maxwell equations important? ›

In 1865 Maxwell wrote down an equation to describe these electromagnetic waves. The equation showed that different wavelengths of light appear to us as different colours. But more importantly, it revealed that there was a whole spectrum of invisible waves, of which the light we can see was only a small part.

Why are the Maxwell equations important to daily life? ›

Maxwell's equations are sort of a big deal in physics. They're how we can model an electromagnetic wave—also known as light. Oh, it's also how most electric generators work and even electric motors. Essentially, you are using Maxwell's equations right now, even if you don't know it.

Why was Maxwell theory not accepted? ›

One of the reasons why Maxwell's theory was so difficult to follow was due to the development of Maxwell's thought process through different times. This made Maxwell not to identify his physical pictures with reality.

What did Maxwell believe about light? ›

In his formulation of electromagnetism, Maxwell described light as a propagating wave of electric and magnetic fields. More generally, he predicted the existence of electromagnetic radiation: coupled electric and magnetic fields traveling as waves at a speed equal to the known speed of light.

What is Maxwell's hypothesis? ›

According to the Maxwell's EM theory the EM waves propagation contains electric and magnetic field vibration in mutually perpendicular direction. Thus, the changing of electric field give rise to magnetic field.


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