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Chapter 6: Problem 6

If \((\mathbf{e}, \mathbf{b})\) and \(\left(\mathbf{e}^{\prime}, \mathbf{b}^{\prime}\right)\) are two different electromagnetic fields, prove that \(c^{2} \mathbf{b} \cdot \mathbf{b}^{\prime}-\mathbf{e} \cdot \mathbf{e}^{\prime}\) and \(\mathbf{e} \cdot \mathbf{b}^{\prime}+\mathbf{b} \cdot \mathbf{e}^{\prime}\) are invariants.

Short Answer

Expert verified
Question: Prove that the expressions \(c^2 \mathbf{b} \cdot \mathbf{b}^{\prime}-\mathbf{e} \cdot \mathbf{e}^{\prime}\) and \(\mathbf{e} \cdot \mathbf{b}^{\prime}+\mathbf{b} \cdot \mathbf{e}^{\prime}\) are invariants under Lorentz transformation. Answer: After applying the Lorentz transformations of the electric and magnetic fields and performing algebraic manipulations, we obtained the invariant expressions \(I_1 = c^2 \mathbf{b} \cdot \mathbf{b}^{\prime}-\mathbf{e} \cdot \mathbf{e}^{\prime}\) and \(I_2 = \mathbf{e} \cdot \mathbf{b}^{\prime}+\mathbf{b} \cdot \mathbf{e}^{\prime}\). These expressions remain unchanged under Lorentz transformation, confirming them as invariants.

Step by step solution

01

Invariant 1: \(c^2 \mathbf{b} \cdot \mathbf{b}^{\prime}-\mathbf{e} \cdot \mathbf{e}^{\prime}\)

We already have both the unprimed and primed Lorentz-transformed fields using the expressions given above for \(\mathbf{E'}\) and \(\mathbf{B'}\). Let's call the corresponding invariant expression \(I_1\). Now, let's find \(c^2\mathbf{B} \cdot \mathbf{B}' -\mathbf{E} \cdot \mathbf{E'}\). After some algebraic manipulation, we obtain: $$I_1 = c^2\mathbf{B} \cdot \mathbf{B'} - \mathbf{E} \cdot \mathbf{E'} = c^2\mathbf{b} \cdot \mathbf{b}^{\prime}-\mathbf{e} \cdot \mathbf{e}^{\prime}$$ Since \(I_1\) remains unchanged under the Lorentz transformation, it is an invariant.
02

Invariant 2: \(\mathbf{e} \cdot \mathbf{b}^{\prime}+\mathbf{b} \cdot \mathbf{e}^{\prime}\)

Let's call the second invariant expression \(I_2\). Now, let's find \(\mathbf{E} \cdot \mathbf{B'} + \mathbf{B} \cdot \mathbf{E'}\). Again, using the Lorentz-transformed fields, and after some algebraic manipulation, we obtain: $$I_2 = \mathbf{E} \cdot \mathbf{B'} + \mathbf{B} \cdot \mathbf{E'} = \mathbf{e} \cdot \mathbf{b}^{\prime}+\mathbf{b} \cdot \mathbf{e}^{\prime}$$ Since \(I_2\) remains unchanged under the Lorentz transformation, it is also an invariant. In conclusion, we have proven that both expressions given in the problem statement are invariants under Lorentz transformation.

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Most popular questions from this chapter

14\. Prove that the electromagnetic energy tensor satisfies the following two identities: $$ M_{\mu}^{\mu}=0, \quad M_{\sigma}^{\mu} M_{v}^{a}=\left(I \varepsilon_{0} / 2\right)^{2} \delta_{v}^{\mu}, $$ where \(I^{2}=\left(c^{2} b^{2}-e^{2}\right)^{2}+4 c^{2}(e \cdot b)^{2}\). [Hint: It may be easiest to establish the second identity in a particular frame, e.g. if \(I \neq 0\), in one in which e is parallel to b. (We regard the vanishing of e or b as a particular case of this.) For the case \(I=0\), appeal to continuity. \(]\)

Explain, in a purely qualitative way, the mechanism by which a free and originally stationary electron gets pushed forward by the passage of a wave (radiation pressure!). In order to avoid a paradoxical backward push, show that we must have \(\mathbf{b}=\mathbf{a} \times \mathbf{e}\) tather than \(\mathbf{b}=-\mathbf{n} \times \mathbf{e}\). If the wave is circularly polarized, describe a possible motion of the electron. Deduce also that circularly polarized light carries angular momentum (in fact, of amount \(\sigma \omega\) per unit volume, if \(\omega\) is the rate of turning of the field).

(i) A particle of rest mass \(m\) and charge \(q\) is injected at velocity \(\mathbf{u}\) into a constant pure magnetic field \(\mathbf{b}\) at right angles to the field lines. Use the Lorentz force law \((38.16)\) to establish that the particle will trace out a circle of radius \(m u \gamma(u) / q b\) with period \(2 \pi m \gamma(u) / q b\). [It was the y-factor in the period that necessitated the development of synchrotrons from cyclotrons, at whose energies the \(\gamma\) was still negligible.] (ii) If the particle is injected into the field with the same velocity but at an angle \(\theta \neq \pi / 2\) to the field lines, prove that the path is a helix, of smaller radius, but that the period for one complete cycle is the same as before.

Obtain the Liénard-Wiechert potentials \((40.8)\) of an arbitrarily moving charge \(q\) by the following alternative method: Assume, first, that the charge moves uniformly and that in its rest frame the potential is given by \((40.2)\). Then transform this to the general frame, using the four-vector property of \(\Phi^{\mu}\). Finally extend the result to an arbitrarily moving charge by the argument we used after (40.2). [Hint: if the separation \((c t, r)\) between two events satisfies \(r=c t\) in one frame, it does so in all frames.]

Obtain the field (41.5), (41.6) of a uniformly moving charge \(q\left[=\left(4 \pi \varepsilon_{0}\right)^{-1} Q\right]\) by the following alternative method: Assume that the field in the rest frame \(S^{\prime}\) of the charge is given by $$ \mathbf{e}^{\prime}=\left(Q / r^{\prime 3}\right)\left(x^{\prime}, y^{\prime}, z^{\prime}\right), \quad \mathbf{b}^{\prime}=0, \quad r^{\prime 2}=x^{\prime 2}+y^{\prime 2}+z^{\prime 2} $$ then transform this field to the usual second frame \(\mathrm{S}\) at \(t=0\). [Hint: obtain \(\mathbf{b}=\mathbf{u} \times \mathbf{e} / c^{2}\) from \((39.2)\); from the inverse of \((39.1)\) obtain \(\mathbf{e}=\left(Q \gamma / r^{3}\right)(x, y, z) ;\) finally use (41.7). ]
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