Page 354 - Intro to Tensor Calculus

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I 8. A homogeneous dielectric is defined by D i and E i having parallel vector fields. Show that for a
j
homogeneous dielectric =0.
i,k
I 9. Show that for a homogeneous, isotropic dielectric medium that is a constant.

I 10. Show that for a homogeneous, isotropic linear dielectric in Cartesian coordinates

α e
P i,i = ρ f .
1+ α e
I 11. Verify the Maxwell’s equations in Gaussian units for a charge free isotropic homogeneous dielectric.
~
~
1 ~ 1 ∂B µ ∂H
~
~
∇· E = ∇· D =0 ∇× E = − c ∂t = − c ∂t

~
~
~
~
∇· B =µ∇H =0 ~ 1 ∂D 4π ~ ∂E 4π ~
∇× H = + J = + σE
c ∂t c c ∂t c
I 12. Verify the Maxwell’s equations in Gaussian units for an isotropic homogeneous dielectric with a
charge.
~
1 ∂B
~
~
∇· D =4πρ ∇× E = −
c ∂t
~
~
∇· B =0 ~ 4π ~ 1 ∂D
∇× H = J +
c c ∂t
I 13. For a volume charge ρ in an element of volume dτ located at a point (ξ, η, ζ) Coulombs law is
ZZZ
1 ρ
~
E(x, y, z)= 2 b e r dτ
4π 0 V r
2
2
2
2
(a) Show that r =(x − ξ) +(y − η) +(z − ζ) .
1
(b) Show that b e r = ((x − ξ) b e 1 +(y − η) b e 2 +(z − ζ) b e 3 ) .
r
(c) Show that
ZZZ ZZZ
1 (x − ξ) b e 1 +(y − η) b e 2 +(z − ζ) b e 3 1 b e r
~
E(x, y, z)= 2 2 2 3/2 ρdξdηdζ = ∇ 2 ρdξdηdζ
4π 0 [(x − ξ) +(y − η) +(z − ζ) ] 4π 0 r
V ZZZ V
1 ρ(ξ, η, ζ)
~
(d) Show that the potential function for E is V = 2 2 2 1/2 dξdηdζ
4π 0 V [(x − ξ) +(y − η) +(z − ζ) ]
~
(e) Show that E = −∇V.
ρ
2
(f) Show that ∇ V = − Hint: Note that the integrand is zero everywhere except at the point where

(ξ, η, ζ)=(x, y, z). Consider the integral split into two regions. One region being a small sphere
about the point (x, y, z) in the limit as the radius of this sphere approaches zero. Observe the identity

b e r b e r
∇ (x,y,z) 2 = −∇(ξ, η, ζ) 2 enables one to employ the Gauss divergence theorem to obtain a
r r
ZZ
ρ b e r ρ
surface integral. Use a mean value theorem to show − 2 · ˆndS = 4π since ˆn = − b e r .
4π 0 S r 4π 0
∗
I 14. Show that for a point charge in space ρ = qδ(x − x 0 )δ(y − y 0 )δ(z − z 0 ), where δ is the Dirac delta
function, the equation (2.6.5) can be reduced to the equation (2.6.1).
I 15.
~
1
(a) Show the electric field E = r 2 b e r is irrotational. Here b e r = r ~ r is a unit vector in the direction of r.
~
(b) Find the potential function V such that E = −∇V which satisfies V(r 0 )= 0 for r 0 > 0.