SYDE 211
Problem Set 3
1. Use the Divergence (Gauss’) Theorem to solve the following problems from the last
problem set:
(a) Suppose T (x, y, z) = x2 + y 2 + z 2 represents the temperature in a region of space
containing the origin of the coordinate system. Compute the heat flux across the
unit sphere.
(b) Consider the cube {(x, y, z) | |x| ≤ 1, |y| ≤ 1, |z| ≤ 1}. Find the charge enclosed
by the cube if the electric field is:
(i) E(x, y, z) = (x, y, z)
(ii) E(x, y, z) = (x2 , y 2 , z 2 )
(c) Let F(x, y, z) = (xz, yz, x2 + y 2 ). Find the flux of F across the boundary surface
of the solid given by x2 + y 2 ≤ z ≤ 1.
2. Again from the last problem set. Consider the cylinder x2 + z 2 = b2 , −ℓ ≤ y ≤ ℓ.
(a) Let a fluid˚
flow have velocity v = (0, 0, kz), k constant, and constant density ρ0 .
∇ · F dV , where F is the mass flux vector and Ω is the region inside
Compute
Ω
the cylinder.
˚
(b) Compute
∇ · F dV , this time with v = (kx, ky, kz).
Ω
(c) Did you get the same answers as those in the last assignment for the flux? Explain
why or why not, and verify the Divergence Theorem by completing any necessary
calculations.
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3. Gauss’ Law
The electric field due to a point charge q at the origin is given by:
E(r) =
where r = (x, y, z), r = ∥r∥, and q =
q
r
r3
Q
is constant.
4πϵ0
Show that
¨
E · n dS =



0
if S does not enclose the charge
Q



if S encloses the charge
ϵ0
where S is a smooth, closed surface. For the second case, you may assume that S is a
S
sphere centered at the origin.
4. For each law below, write the law mathematically in terms of integrals. Then use
theorems from vector calculus to derive a partial differential equation that holds at
every point in space, each leading to one of Maxwell’s equations.
(a) Faraday’s Law: the circulation of an electric field E around the perimeter of a
surface is equal to the negative time rate of change of the flux of the magnetic
field B through the surface.
(b) The flux of the magnetic field through any closed surface is zero.
(c) Ampere’s Law: The circulation of the magnetic field around the perimeter of a
surface is equal to the time rate of change of the flux of the electric field through
the surface + the flux of the electric current density through the surface (where
there are constants of proportionality that you can look up).
5. Use Stokes’ Theorem to compute the circulation of F = (xy, yz, zx) around the curve C
which is the triangle with vertices (1, 0, 0), (0, 1, 0), (0, 0, 1), oriented counterclockwise
as viewed from above. (On which surface does C lie?)
2
¨
(∇ × F) · n dS,
6. (a) Use Stokes’ Theorem to evaluate
Σ
where F = (y − z, −x − z, x + y), Σ is the portion of z = 9 − x2 − y 2 with z ≥ 0
and n is the upward pointing unit normal.
(b) Let S be the (plane) surface x2 + y 2 ≤ 9, z =¨0, with n pointing in the positive
(∇×F)·n dS from the definition.
z−direction, and let F be as above. Calculate
S
(c) What is the connection between (a) and (b)?
7. Find the flux of the vector field F = (xz, −yz, 1 + y 2 ) across the surface Σ given by
Σ = {(x, y, z) | z = cos−1 (x2 + y 2 ), x2 + y 2 ≤ 1} with outward pointing normal.
Hint: Do not attempt to do this directly. Instead, notice that ∇ · F = 0 which implies
there is a vector potential, call it G, such that F = ∇ × G. Also, note that vector
potentials are not unique, and therefore you can choose any G that satisfies the above.
Start with the F1 and F2 components, and see if you can find G3 that works.
8. Maxwell’s equations are:
∇·E=
ρ
ϵ0
∇×E=−
(0.1)
∂B
∂t
∇·B=0
∇ × B = µ0 ϵ0
(0.2)
(0.3)
∂E
+ µ0 J
∂t
(0.4)
where E and B are the electric and magnetic fields respectively, ρ is the charge density,
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J is the current density, and ϵ0 , µ0 are constants, with c = √
the speed of light
µ0 ϵ0
in a vacuum. Show that in the case of no charges and no currents, the E field satisfies
the wave equation
∂ 2E
− c2 ∇ 2 E = 0
2
∂t
Use the identity: ∇ × (∇ × E) = ∇(∇ · E) − ∇2 E.
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