Page 355 - Intro to Tensor Calculus

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I 16.
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(a) If E is a conservative electric field such that E = −∇V, then show that E is irrotational and satisfies
~
~
∇× E =curl E =0.
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~
~
~
(b) If ∇× E =curl E = 0, show that E is conservative. (i.e. Show E = −∇V.)
Hint: The work done on a test charge Q = 1 along the straight line segments from (x 0 ,y 0 ,z 0 )to
(x, y 0 ,z 0) and then from (x, y 0 ,z 0)to (x, y, z 0 ) and finally from (x, y, z 0 )to(x, y, z) can be written
Z x Z y Z z
V = V(x, y, z)= − E 1 (x, y 0 ,z 0) dx − E 2 (x, y, z 0 ) dy − E 3 (x, y, z) dz.
x 0 y 0 z 0
Now note that
Z z
∂V ∂E 3 (x, y, z)
= −E 2 (x, y, z 0 ) − dz
∂y ∂y
z 0
~
and from ∇×E = 0 we find ∂E 3 = ∂E 2 , which implies ∂V = −E 2 (x, y, z). Similar results are obtained
∂y ∂z ∂y
∂V ∂V
~
for and . Hence show −∇V = E.
∂x ∂z
I 17.
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(a) Show that if ∇· B = 0, then there exists some vector field A such that B = ∇× A.
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The vector field A is called the vector potential of B.
Z
1
~
~
r
Hint: Let A(x, y, z)= sB(sx, sy, sz) × ~rds where ~ = x b e 1 + y b e 2 + z b e 3
0
1
Z
dB i 2
and integrate s ds by parts.
0 ds
~
(b) Show that ∇· (∇× A)= 0.
I 18. Use Faraday’s law and Ampere’s law to show
i
∂ i ∂E
jm
i
im
j
g (E ) ,m − g E = −µ 0 J + 0
,j ,mj
∂t ∂t
~
~
I 19. Assume that J = σE where σ is the conductivity. Show that for ρ = 0 Maxwell’s equations produce
~
2 ~
∂E ∂ E 2 ~
µ 0 σ + µ 0 0 =∇ E
∂t ∂t 2
2 ~
~
∂B ∂ B 2 ~
and µ 0 σ + µ 0 0 2 =∇ B.
∂t ∂t
~
~
Here both E and B satisfy the same equation which is known as the telegrapher’s equation.
I 20. Show that Maxwell’s equations (2.6.75) through (2.6.78) for the electric field under electrostatic
conditions reduce to
~
∇× E =0
~
∇· D =ρ f
~ ~ 2 ρ f
Now E is irrotational so that E = −∇V. Show that ∇ V = − .