Irodov 3.355: What Maxwell's Equations Forbid – Amit & Deepali Sinha Foundation

Problem Irodov 3.355

Use Maxwell’s equations to show that:

a magnetic field that changes in time cannot exist without an electric field;
a uniform electric field cannot coexist with a magnetic field that changes in time;
inside an empty cavity a uniform electric field (or a uniform magnetic field) can depend on time.

Given

Maxwell’s equations
(c) no charges or currents in the cavity

Find

show (a), (b), (c)

Solution

Step 1The two curl equations

Each part turns on the two Maxwell equations that tie the change of one field in time to the variation of the other in space:

\begin{align*} \nabla\times\vect E &= -\frac{\partial\vect B}{\partial t} \tag{1} \\ \nabla\times\vect H &= \vect j + \frac{\partial\vect D}{\partial t} \tag{2}\end{align*}

Step 2(a) A changing B needs an electric field

If $\partial\vect B/\partial t \neq 0$ somewhere, then by (1) the curl of $\vect E$ is not zero there. A curl is built from spatial derivatives of $\vect E$ , and these can differ from zero only if $\vect E$ itself is not zero in that neighbourhood:

\frac{\partial\vect B}{\partial t} \neq 0 \;\Longrightarrow\; \nabla\times\vect E \neq 0 \;\Longrightarrow\; \vect E \neq 0

(3)

A region where B, out of the page, grows in time: it is ringed by closed lines of electric field, inside the region and outside it. — A region where $\vect B$ , out of the page, grows in time: it is ringed by closed lines of electric field, inside the region and outside it.

Step 3(b) A uniform E forbids a changing B

A uniform field has the same value everywhere, so every spatial derivative of $\vect E$ vanishes and so does its curl. Then (1) fixes the magnetic field’s rate of change:

\vect E\ \text{uniform} \;\Longrightarrow\; \frac{\partial\vect B}{\partial t} = -\nabla\times\vect E = 0

(4)

A time-dependent $\vect B$ would contradict it, so the two cannot coexist.

Step 4(c) A uniform E that changes in time

In an empty cavity $\vect j = 0$ and $\vect D = \varepsilon_0\vect E$ . Let the field be uniform and changing, $\vect E = \vect a\,f(t)$ with $\vect a$ a constant vector. By (4) the magnetic field cannot change in time. But (2) demands a curl of $\vect H$ :

\nabla\times\vect H = \varepsilon_0\,\vect a\,\dot f(t)

Whatever $\vect H$ solves this, it may not change in time, since $\vect B = \mu_0\vect H$ in the cavity and (4) holds. Differentiating the equation in time therefore leaves zero on the left:

\nabla\times\frac{\partial\vect H}{\partial t} = \varepsilon_0\,\vect a\,\ddot f(t) = 0

So $\ddot f = 0$ and $f$ is linear in $t$ :

\vect E = \vect a\,(\alpha + \beta t)

(5)

with $\alpha$ and $\beta$ constants. Such a field does exist: with $\dot f = \beta$ , a magnetic field circulating round the direction of $\vect a$ , growing linearly with the distance, with $\vect r$ measured from any point of the cavity, is

\vect H = \frac{\varepsilon_0\beta}{2}\,\vect a\times\vect r

It is steady, and its curl is $\varepsilon_0\beta\,\vect a$ , as required, by the identity for a constant vector $\vect c$ :

\nabla\times(\vect c\times\vect r) = 2\vect c

So a uniform field growing at a steady rate is allowed, and it carries this steady magnetic field. The same argument with the roles exchanged allows a uniform magnetic field growing at a steady rate, $\vect b\,(\alpha + \beta t)$ : (1) is then satisfied by a steady circulating electric field,

\vect B = \vect b\,(\alpha + \beta t), \qquad \vect E = -\frac{\beta}{2}\,\vect b\times\vect r

and since this $\vect E$ does not change, (2) holds with $\nabla\times\vect H = 0$ , which a uniform $\vect H$ satisfies.

Check

The constructed pair in (c) satisfies all four equations: $\nabla\cdot\vect E = 0$ (uniform), $\nabla\cdot\vect B = 0$ (the divergence of $\vect a\times\vect r$ is zero), $\nabla\times\vect E = 0 = -\partial\vect B/\partial t$ when $\dot f$ is constant, and $\nabla\times\vect H = \varepsilon_0\beta\,\vect a = \partial\vect D/\partial t$ . With $\ddot f \neq 0$ the second last fails, which is exactly the case (c) excludes.

Remark

The steadily charging parallel-plate capacitor is the everyday example of (c): between its plates $\vect E$ is uniform and grows linearly while the charging current is constant, and the magnetic field circling the axis, proportional to the distance from it, is the steady field of the displacement current.