[The naive explanation for the pressure of light offered in Section 9.2.3 has its flaws, as you discovered if you worked Problem 9.11. Here’s another account, due originally to Planck.^25] A plane wave traveling through vacuum in the z direction encounters a perfect conductor occupying the reg

Question

[The naive explanation for the pressure of light offered in Section 9.2.3 has its flaws, as you discovered if you worked Problem 9.11. Here’s another account, due originally to Planc[latex]k \cdot^{2}[/latex]] A plane wave traveling through vacuum in the z direction encounters a perfect conductor occupying the region z ≥ 0, and reflects back:

[latex]E (z, t)=E_{0}[\cos (k z-\omega t)-\cos (k z+\omega t)] \hat{ x }, \quad(z<0)[/latex].

(a) Find the accompanying magnetic field (in the region z < 0).

(b) Assuming B = 0 inside the conductor, find the current K on the surface z = 0, by invoking the appropriate boundary condition.

(c) Find the magnetic force per unit area on the surface, and compare its time average with the expected radiation pressure (Eq. 9.64)

[latex]P=\frac{1}{A} \frac{\Delta p}{\Delta t}=\frac{1}{2} \epsilon_{0} E_{0}^{2}=\frac{I}{c}[/latex] (9.64)

Accepted Answer

(a) Since (E × B) points in the direction of propagation, [latex]B =\frac{E_{0}}{c}[\cos (k z-\omega t)+\cos (k z+\omega t)] \hat{ y } [/latex] .

(b) From Eq. 7.63, [latex]K imes(-\hat{ z })=\frac{1}{\mu_{0}} B =\frac{E_{0}}{\mu_{0} c}[2 \cos (\omega t)] \hat{ y }, K =\frac{2 E_{0}}{\mu_{0} c} \cos (\omega t) \hat{ x }[/latex].

[latex]H _{1}^{\|}- H _{2}^{\|}= K _{f} imes \hat{ n }[/latex] (7.63)

(c) The force per unit area is [latex]f = K imes B _{ ext {ave }}=\frac{2 E_{0}^{2}}{\mu_{0} c^{2}}[\cos (\omega t) \hat{ x }] imes[\cos (\omega t) \hat{ y }]=2 \epsilon_{0} E_{0}^{2} \cos ^{2}(\omega t) \hat{ z }[/latex] . The time average of [latex]\cos ^{2}(\omega t)[/latex] is 1/2, so

[latex]f _{ ext {ave }}=\epsilon_{0} E_{0}^{2}[/latex].

This is twice the pressure in Eq. 9.64, but that was for a perfect absorber, whereas this is a perfect reflector.

Question 9.34: [The naive explanation for the pressure of light offered in ...

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