Maxwell's Equations

Main idea

Four equations hold all of electromagnetism. Electric field lines start and end on charges, magnetic field lines never end, a changing magnetic field swirls the electric field, and currents and changing electric fields swirl the magnetic field. Each one is a statement about flux or circulation over a whole surface or loop (the integral form), or, shrunk down to a single point, about divergence or curl (the differential form).

Gauss's law

\begin{aligned}\oiint_S \mathbf{E}\cdot d\mathbf{A} &= \frac{{\htmlData{sym=0}{Q_{\text{enc}}}}}{\varepsilon_0}\\ {\htmlData{sym=2}{\nabla\cdot\mathbf{E}}} &= \frac{\htmlData{sym=1}{{\rho}}}{\varepsilon_0}\end{aligned}

What the symbols mean

$Q_{\text{enc}}$
Enclosed charge
The total charge inside the closed surface S.
$\rho$
Charge density
Charge per unit volume at a point, in C/m³.
$\nabla\cdot\mathbf{E}$
Divergence of E
How much E spreads out from a point: the outward flux of a tiny box, per unit volume.

Gauss's law for magnetism

\begin{aligned}\oiint_S \mathbf{B}\cdot d\mathbf{A} &= 0\\ {\htmlData{sym=0}{\nabla\cdot\mathbf{B}}} &= 0\end{aligned}

What the symbols mean

$\nabla\cdot\mathbf{B}$
Divergence of B
Zero everywhere: no point is a source or a sink of magnetic field.

Faraday's law

\begin{aligned}\oint_C \mathbf{E}\cdot d\boldsymbol{\ell} &= -\frac{d{\htmlData{sym=0}{\Phi_B}}}{dt}\\ {\htmlData{sym=1}{\nabla\times\mathbf{E}}} &= -\frac{\partial\mathbf{B}}{\partial t}\end{aligned}

What the symbols mean

$\Phi_B$
Magnetic flux
The flux of B through a surface whose edge is the loop C.
$\nabla\times\mathbf{E}$
Curl of E
How much E circulates around a point, per unit area.

Ampère–Maxwell law

\begin{aligned}\oint_C \mathbf{B}\cdot d\boldsymbol{\ell} &= \mu_0{\htmlData{sym=0}{I_{\text{enc}}}} + \mu_0\varepsilon_0\frac{d{\htmlData{sym=1}{\Phi_E}}}{dt}\\ \nabla\times\mathbf{B} &= \mu_0{\htmlData{sym=2}{\mathbf{J}}} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}\end{aligned}

What the symbols mean

$I_{\text{enc}}$
Enclosed current
Conduction current through a surface whose edge is C.
$\Phi_E$
Electric flux
Its rate of change, times ε₀, is the displacement current.
$\mathbf{J}$
Current density
Current per unit area at a point, in A/m².

Lorentz force: how the fields push on charges

\mathbf{F} = {\htmlData{sym=0}{q}}\left(\mathbf{E} + {\htmlData{sym=1}{\mathbf{v}}}\times\mathbf{B}\right)

What the symbols mean

$q$
Charge
The charge of the particle the fields act on.
$\mathbf{v}$
Velocity
The particle's velocity.

Explanation

Maxwell's equations are simply four rules that describe everything electric and magnetic: where electric fields come from, why magnetic field lines never end, and how a changing field of one kind makes a field of the other kind. They're named after James Clerk Maxwell, who pulled these laws together in the 1860s and added the missing displacement current (the tidy four-equation form came a bit later, from Oliver Heaviside). Every antenna, motor, transformer and fiber-optic cable is designed from them. The symbols are just a few pieces: $\nabla\cdot$ (divergence, how much a field spreads out from a point), $\nabla\times$ (curl, how much it swirls around a point), $\partial/\partial t$ (how fast something changes in time) and $\rho$ and $\mathbf{J}$ (the charges and currents). Four lines full of upside-down triangles look like the hardest thing in the course, but each one is a sentence you've already met in an earlier unit. Bear with it.

One page of physics. Everything in this course so far, from Coulomb's law to LC circuits, follows from four equations about two fields, plus the Lorentz force that says how those fields push on charges. It helps to read them as two pairs.

The divergence pair: where field lines begin.

Gauss's law. The flux of $\mathbf{E}$ out of any closed surface counts the charge inside. Electric field lines start on positive charge and end on negative charge (Gauss's law).
Gauss's law for magnetism. The flux of $\mathbf{B}$ out of any closed surface is zero. There are no magnetic charges, so magnetic field lines never start or stop: they close on themselves. Cut a bar magnet in half and you get two shorter magnets, never a lone north pole.

The curl pair: what makes fields circulate.

Faraday's law. A changing magnetic flux drives a circulating $\mathbf{E}$ , the EMF in every generator and transformer (Faraday's law). The minus sign is Lenz's law.
Ampère–Maxwell law. Currents and changing electric flux drive a circulating $\mathbf{B}$ (Ampère's law plus the displacement current).

Integral form ⟺ differential form. The integral forms talk about whole surfaces and loops; the differential forms talk about single points. The divergence theorem converts one into the other. For any small volume $V$ with surface $S$ ,

\begin{gathered}\oiint_S\mathbf{E}\cdot d\mathbf{A} = \iiint_V\nabla\cdot\mathbf{E}\,dV,\\ \frac{Q_{\text{enc}}}{\varepsilon_0} = \iiint_V\frac{\rho}{\varepsilon_0}\,dV.\end{gathered}

Gauss's law says the left sides are equal for every volume, so the integrands must match point by point: $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ . (If they differed at some point, a tiny volume around it would break the law.) Stokes' theorem does the same for the loop laws. For Faraday's law on a fixed surface $S$ with edge $C$ ,

\begin{gathered}\oint_C\mathbf{E}\cdot d\boldsymbol{\ell} = \iint_S(\nabla\times\mathbf{E})\cdot d\mathbf{A},\\ -\frac{d}{dt}\iint_S\mathbf{B}\cdot d\mathbf{A} = \iint_S\left(-\frac{\partial\mathbf{B}}{\partial t}\right)\cdot d\mathbf{A},\end{gathered}

and the left sides are equal for every surface, so $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ . Use the integral form when symmetry lets you pull $E$ or $B$ out of the integral (spheres, cylinders, circles), and the differential form to check a proposed field or to find the charges and currents that made it.

Charge conservation is built in. Take the divergence of the Ampère–Maxwell law. The divergence of a curl is zero and $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ , so

0 = \mu_0\nabla\cdot\mathbf{J} + \mu_0\varepsilon_0\frac{\partial}{\partial t}\nabla\cdot\mathbf{E} = \mu_0\left(\nabla\cdot\mathbf{J} + \frac{\partial\rho}{\partial t}\right).

Charge can't appear or vanish, only flow. Notice that this only worked because of Maxwell's term. Without it, the equations would contradict charge conservation, which was exactly the puzzle in the last lesson.

The static special cases. When nothing changes, Faraday's law says $\nabla\times\mathbf{E} = \mathbf{0}$ , so $\mathbf{E} = -\nabla V$ is a gradient (field from potential), and Gauss's law becomes Poisson's equation $\nabla^2V = -\rho/\varepsilon_0$ (Poisson's and Laplace's equations). Electrostatics and magnetostatics are what's left when the time derivatives vanish. The time derivatives are what couple $\mathbf{E}$ and $\mathbf{B}$ into one electromagnetic field.

Empty space. With $\rho = 0$ and $\mathbf{J} = \mathbf{0}$ the equations are nearly symmetric:

\begin{gathered}\nabla\cdot\mathbf{E} = 0, \qquad \nabla\cdot\mathbf{B} = 0,\\ \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{B} = \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}.\end{gathered}

A changing $\mathbf{B}$ makes $\mathbf{E}$ and a changing $\mathbf{E}$ makes $\mathbf{B}$ . So what happens if each keeps making the other? The next lesson shows they can keep each other going forever, and that's light.

A common mistake. After a whole course of electrostatics, it's tempting to assume $\nabla\times\mathbf{E} = \mathbf{0}$ always, so that every $\mathbf{E}$ is the gradient of some potential. That's only true when $\mathbf{B}$ is steady. Around a transformer, $\mathbf{E}$ circulates and isn't the gradient of any potential. Two smaller slips: reading $\nabla\cdot\mathbf{B} = 0$ as " $\mathbf{B} = 0$ " (it only rules out sources and sinks), and mismatching orientations. The loop's direction and the surface's normal must follow the right-hand rule, or Faraday's minus sign flips.

See it

This steps through the four laws one at a time. First, the electric field of a + and a − charge runs from one to the other. Second, around a loop of current seen edge-on, the magnetic field lines close on themselves, because nothing is a magnetic charge. Third, a growing B out of the page (a solenoid seen end-on) drives E clockwise around it, the EMF a transformer uses. Fourth, a growing E out of the page (a charging capacitor seen face-on) drives B counterclockwise. Notice the last two are the same picture turning opposite ways; that's the minus sign in Faraday's law.

This is a long solenoid seen end-on. Inside the dashed circle (R = 2 cm) its field points out of the page and grows at 100 T/s, so Faraday's law drives E clockwise around it (the arrows show direction only). As the loop grows, the EMF rises as πr²(dB/dt), then holds at 126 mV once the loop encloses all the flux. Notice that E still circulates outside the solenoid, even though its curl is zero there: all the curl is inside, and the integral form just adds it up. That's how a transformer's secondary coil picks up voltage from a core it never touches.

Worked examples

Example 1

The field inside a long solenoid of radius $R = 3.0$ cm points along $+z$ and grows at $dB/dt = 50$ T/s. (a) Use Faraday's law in integral form to find the induced electric field $2.0$ cm and $5.0$ cm from the axis. (b) Check the differential form.

1
By symmetry, $\mathbf{E}$ circles the axis with the same strength all the way around a circle of radius $r$ , so $\oint\mathbf{E}\cdot d\boldsymbol{\ell} = E\cdot2\pi r$ .

Example 2

In a region of space the fields are steady: $\mathbf{E} = (200\text{ V/m}^2)\langle x, y, 0\rangle$ and $\mathbf{B} = (3.0\ \mu\text{T/m})\langle -y, x, 0\rangle$ , with $x$ and $y$ in meters. What charge density and current density produce them?

1
Gauss's law: $\nabla\cdot\mathbf{E} = 200 + 200 = 400$ V/m², so $\rho = \varepsilon_0(400) = 3.54\times10^{-9}$ C/m³, the same everywhere.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Which of Maxwell's equations rules out magnetic charges, so that magnetic field lines never begin or end?

Problem 2+5 XP

For what value of $a$ is $\mathbf{B} = B_0\langle 3x, ay, z\rangle/L$ a possible magnetic field? ( $B_0$ and $L$ are constants.)

Problem 3+5 XP

A region holds the electric field $\mathbf{E} = (5.0\text{ V/m}^2)\langle y, -x, 0\rangle$ , with $x$ and $y$ in meters. Use Faraday's law to find $\partial B_z/\partial t$ , in T/s.

Problem 4+5 XP

In a region of space, $\mathbf{E} = \langle 4000x^2,\ 4000y,\ 4000z\rangle$ V/m, with $x$ , $y$ and $z$ in meters. Find the charge density at the point $(0.50, 0, 0)$ m, in nC/m³, to 3 significant figures.

Problem 5+5 XP

A magnetic stimulation coil held against the head makes a field that rises steadily from $0$ to $1.5$ T in $100\ \mu$ s. Model the field as uniform over a circle of radius $3.0$ cm and zero outside it. What electric field does it induce $2.0$ cm from the axis, in V/m?

Problem 6+5 XP

A circular loop of radius $10$ cm lies in the $xy$ -plane in a uniform field $\mathbf{B} = B(t)\,\hat{\mathbf{z}}$ that changes at $dB/dt = -0.40$ T/s. Find $\oint\mathbf{E}\cdot d\boldsymbol{\ell}$ around the loop, counterclockwise as seen from above ( $+z$ ), in mV.

Problem 7+5 XP

In a region where nothing changes with time, which equation guarantees that the electric field can be written as $\mathbf{E} = -\nabla V$ ?

Problem 8+5 XP

In part of a device, the current density is $\mathbf{J} = (2.0\times10^3\text{ A/m}^4)\,x^2\,\hat{\mathbf{x}}$ , with $x$ in meters. At $x = 5.0$ cm, how fast is the charge density changing, in C/(m³·s)? (Use the continuity equation that Maxwell's equations imply.)

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

An electron beam of radius $1.0$ mm carries a current of $2.0$ mA, its electrons moving at $v = 1.0\times10^8$ m/s. Treat it as a long, uniform cylinder of charge. (a) Use Gauss's law to find the electric field at the beam's edge. (b) Use Ampère's law to find the magnetic field there. (c) An electron at the edge feels an outward electric push and an inward magnetic pull. What percentage of the push does the pull cancel? Enter $E$ (V/m), $B$ (nT) and the percentage, to 3 significant figures.

Where it's used

Maxwell's equations are what electromagnetic simulation software solves. Engineers designing antennas, phone circuit boards, MRI coils, microwave ovens and radar discretize the differential forms on a grid (finite-difference time-domain and finite-element codes) and step them forward in time. The integral forms are the quick hand checks: Gauss's law for the field near a charged conductor, Ampère's law for the field around a busbar, Faraday's law for the voltage a transformer winding picks up.

Total Charge on a PlateCalc III · Electrical Engineering The Magnetic Field Around a WireCalc III · Electrical Engineering Electric Fields and EquipotentialsCalc III · Physics

The Displacement Current Electromagnetic Waves

Maxwell's Equations

Main idea

Gauss's law

\begin{aligned}\oiint_S \mathbf{E}\cdot d\mathbf{A} &= \frac{{\htmlData{sym=0}{Q_{\text{enc}}}}}{\varepsilon_0}\\ {\htmlData{sym=2}{\nabla\cdot\mathbf{E}}} &= \frac{\htmlData{sym=1}{{\rho}}}{\varepsilon_0}\end{aligned}

What the symbols mean

$Q_{\text{enc}}$
Enclosed charge
The total charge inside the closed surface S.
$\rho$
Charge density
Charge per unit volume at a point, in C/m³.
$\nabla\cdot\mathbf{E}$
Divergence of E
How much E spreads out from a point: the outward flux of a tiny box, per unit volume.

Gauss's law for magnetism

\begin{aligned}\oiint_S \mathbf{B}\cdot d\mathbf{A} &= 0\\ {\htmlData{sym=0}{\nabla\cdot\mathbf{B}}} &= 0\end{aligned}

What the symbols mean

$\nabla\cdot\mathbf{B}$
Divergence of B
Zero everywhere: no point is a source or a sink of magnetic field.

Faraday's law

\begin{aligned}\oint_C \mathbf{E}\cdot d\boldsymbol{\ell} &= -\frac{d{\htmlData{sym=0}{\Phi_B}}}{dt}\\ {\htmlData{sym=1}{\nabla\times\mathbf{E}}} &= -\frac{\partial\mathbf{B}}{\partial t}\end{aligned}

What the symbols mean

$\Phi_B$
Magnetic flux
The flux of B through a surface whose edge is the loop C.
$\nabla\times\mathbf{E}$
Curl of E
How much E circulates around a point, per unit area.

Ampère–Maxwell law

\begin{aligned}\oint_C \mathbf{B}\cdot d\boldsymbol{\ell} &= \mu_0{\htmlData{sym=0}{I_{\text{enc}}}} + \mu_0\varepsilon_0\frac{d{\htmlData{sym=1}{\Phi_E}}}{dt}\\ \nabla\times\mathbf{B} &= \mu_0{\htmlData{sym=2}{\mathbf{J}}} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}\end{aligned}

What the symbols mean

$I_{\text{enc}}$
Enclosed current
Conduction current through a surface whose edge is C.
$\Phi_E$
Electric flux
Its rate of change, times ε₀, is the displacement current.
$\mathbf{J}$
Current density
Current per unit area at a point, in A/m².

Lorentz force: how the fields push on charges

\mathbf{F} = {\htmlData{sym=0}{q}}\left(\mathbf{E} + {\htmlData{sym=1}{\mathbf{v}}}\times\mathbf{B}\right)

What the symbols mean

$q$
Charge
The charge of the particle the fields act on.
$\mathbf{v}$
Velocity
The particle's velocity.

Explanation

The divergence pair: where field lines begin.

Gauss's law. The flux of $\mathbf{E}$ out of any closed surface counts the charge inside. Electric field lines start on positive charge and end on negative charge (Gauss's law).
Gauss's law for magnetism. The flux of $\mathbf{B}$ out of any closed surface is zero. There are no magnetic charges, so magnetic field lines never start or stop: they close on themselves. Cut a bar magnet in half and you get two shorter magnets, never a lone north pole.

The curl pair: what makes fields circulate.

Faraday's law. A changing magnetic flux drives a circulating $\mathbf{E}$ , the EMF in every generator and transformer (Faraday's law). The minus sign is Lenz's law.
Ampère–Maxwell law. Currents and changing electric flux drive a circulating $\mathbf{B}$ (Ampère's law plus the displacement current).

\begin{gathered}\oiint_S\mathbf{E}\cdot d\mathbf{A} = \iiint_V\nabla\cdot\mathbf{E}\,dV,\\ \frac{Q_{\text{enc}}}{\varepsilon_0} = \iiint_V\frac{\rho}{\varepsilon_0}\,dV.\end{gathered}

\begin{gathered}\oint_C\mathbf{E}\cdot d\boldsymbol{\ell} = \iint_S(\nabla\times\mathbf{E})\cdot d\mathbf{A},\\ -\frac{d}{dt}\iint_S\mathbf{B}\cdot d\mathbf{A} = \iint_S\left(-\frac{\partial\mathbf{B}}{\partial t}\right)\cdot d\mathbf{A},\end{gathered}

Charge conservation is built in. Take the divergence of the Ampère–Maxwell law. The divergence of a curl is zero and $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ , so

0 = \mu_0\nabla\cdot\mathbf{J} + \mu_0\varepsilon_0\frac{\partial}{\partial t}\nabla\cdot\mathbf{E} = \mu_0\left(\nabla\cdot\mathbf{J} + \frac{\partial\rho}{\partial t}\right).

Empty space. With $\rho = 0$ and $\mathbf{J} = \mathbf{0}$ the equations are nearly symmetric:

\begin{gathered}\nabla\cdot\mathbf{E} = 0, \qquad \nabla\cdot\mathbf{B} = 0,\\ \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{B} = \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}.\end{gathered}

See it

Worked examples

Example 1

1
By symmetry, $\mathbf{E}$ circles the axis with the same strength all the way around a circle of radius $r$ , so $\oint\mathbf{E}\cdot d\boldsymbol{\ell} = E\cdot2\pi r$ .

Example 2

1
Gauss's law: $\nabla\cdot\mathbf{E} = 200 + 200 = 400$ V/m², so $\rho = \varepsilon_0(400) = 3.54\times10^{-9}$ C/m³, the same everywhere.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Which of Maxwell's equations rules out magnetic charges, so that magnetic field lines never begin or end?

Problem 2+5 XP

For what value of $a$ is $\mathbf{B} = B_0\langle 3x, ay, z\rangle/L$ a possible magnetic field? ( $B_0$ and $L$ are constants.)

Problem 3+5 XP

A region holds the electric field $\mathbf{E} = (5.0\text{ V/m}^2)\langle y, -x, 0\rangle$ , with $x$ and $y$ in meters. Use Faraday's law to find $\partial B_z/\partial t$ , in T/s.

Problem 4+5 XP

Problem 5+5 XP

Problem 6+5 XP

Problem 7+5 XP

In a region where nothing changes with time, which equation guarantees that the electric field can be written as $\mathbf{E} = -\nabla V$ ?

Problem 8+5 XP

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Where it's used

Total Charge on a PlateCalc III · Electrical Engineering The Magnetic Field Around a WireCalc III · Electrical Engineering Electric Fields and EquipotentialsCalc III · Physics