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# 10.E: Electromagnetism (Exercises)

### Q1

1. A parallel-plate capacitor has charge per unit area $$±σ$$ on its two plates. Use Gauss’s law to ﬁnd the ﬁeld between the plates.
2.  In the style of Example 10.4.1, transform the ﬁeld to a frame moving perpendicular to the plates, and verify that the result makes sense in terms of the sources that are present.
3. Repeat the analysis for a frame moving parallel to the plates.

### Q2

We’ve seen examples such as ﬁgure 10.1.1 in which a purely magnetic ﬁeld in one frame becomes a mixture of magnetic and electric ﬁelds in another, and also cases like Example 10.4.1 in which a purely electric ﬁeld transforms to a mixture. Can we have a case in which a purely electric ﬁeld in one frame transforms to a purely magnetic one in another? The easy way to do this problem is by using invariants.

### Q3

1. Starting from Equation 10.3.5 for $$\mathcal{F}^{µν}$$, lower an index to ﬁnd $$\mathcal{F}^µ\: _ν$$ . Assume Minkowski coordinates and metric signature $$+---$$.
2.  Let $$v = γ(1,u_x,u_y,u_z)$$, where $$(u_x,u_y,u_z)$$ is the velocity threevector. Write out the matrix multiplication $$F^µ = q\mathcal{F}^µ\:_ν\; v^ν$$, and show that the result is the Lorentz force law.

### Q4

In section 10.6, I presented a list of properties of the electromagnetic stress tensor, followed by an argument in which the tensor is constructed with three unknown constants $$a$$, $$b$$, and $$c$$, to be determined from those properties. The values of $$a$$ and $$b$$ are derived in the text, and the purpose of this problem is to ﬁnish up by proving that $$c = -1$$. The idea is to take the ﬁeld of a point charge, which we know satisﬁes Maxwell’s equations, and then apply property 8, which requires that the energy-conservation condition $$∂T^{ab}/∂x^a = 0$$ hold. This works out nicely if you apply this property to the $$x$$ column of $$T$$, at a point that lies in the positive $$x$$ direction relative to the charge.

### Q5

Show that the number of independent conditions contained in equations  10.7.8 and  10.7.9 agrees with the number found in Maxwell's equations.

### Q6

Show that

$\frac{\partial \mathcal{F} ^{\mu\nu}}{\partial x^\lambda} + \frac{\partial \mathcal{F} ^{\nu\lambda}}{\partial x^\mu} + \frac{\partial \mathcal{F} ^{\lambda\mu}}{\partial x^\nu} = 0$

implies that the magnetic ﬁeld has zero divergence.

### Q7

Write down the ﬁelds of an electromagnetic plane wave propagating in the z direction, choosing some polarization. Do not assume a sinusoidal wave. Show that this is a solution of

$\frac{\partial \mathcal{F} ^{\mu\nu}}{\partial x^\nu} = 0$