   Chapter 16.5, Problem 38E

Chapter
Section
Textbook Problem

Maxwell’s equations relating the electric field E and magnetic field H as they vary with time in a region containing no charge and no current can be stated as follows: div  E  = 0 div  H  = 0 curt  E  = − 1 c ∂ H ∂ t curl  H  = 1 c ∂ E ∂ t where c is the speed of light. Use these equations to prove the following:(a) ∇ × ( ∇ × E ) = − 1 c 2 ∂ 2 E ∂ t 2 (b) ∇ × ( ∇ × H ) = − 1 c 2 ∂ 2 H ∂ t 2 (c) ∇ 2 E = 1 c 2 ∂ 2 E ∂ t 2 [Hint: Use Exercise 29.](d) ∇ 2 H = 1 c 2 ∂ 2 H ∂ t 2

(a)

To determine

To show: The ×(×E)=1c22Et2 .

Explanation

Given data:

divE=0 , divH=0 , curlE=1cHt and curlH=1cEt .

Formula used:

curlF=×F

Consider H=h1,h2,h3 and E=E1,E2,E3 .

Find ×(×E) .

×(×E)=×(curlE)

Substitute 1cHt for curlE ,

×(×E)=×(1cHt)=1c[×(Ht)]=1c[|ijkxyzh1th2th3t|]=1c[(2h3yt2h2zt)i(2h3xt2h1zt)j+(2h2xt2h1yt)k]

Simplify the equation

(b)

To determine

To show: The ×(×H)=1c22Ht2 .

(c)

To determine

To show: The 2E=1c22Et2 .

(d)

To determine

To show: The 2H=1c22Ht2 .

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