MA330: Assignment 4
Required Reading.
• Read Chapter 4, § 42
To be turned in April 12th at the start of class.
1. Recall Maxwell’s equations, as stated in class.
∇·E=
ρ
0
∇·B=0
∇×E=−
∂B
∂t
∇ × B = µ0 J + µ0 0
∂E
∂t
Assume that {µ0 , 0 } are constants.
(a) In class, we showed that the electric field, E, obeys an inhomogeneous wave equation. Using a similar method,
show that the magnetic field, B, also obeys an inhomogeneous wave equation.
(b) Using Maxwell’s equations, derive the following relationship between current density and charge density.
∂ρ
+∇·J=0
∂t
2. Consider the transformation
u=
x
,
x2 + y 2
v=
y
x2 + y 2
and its inverse
u
v
,
y= 2
.
u2 + v 2
u + v2
If the point (x, y) corresponds to (r, θ) in polar coordinates, show that the transformed point (u, v) corresponds
to (1/r, θ) in polar coordinates. In visual terms, this transformation reflects (x, y) across the unit circle.
If f (u, v) is a scalar function, show how to compute grad(f ) in this new coordinate system.
If g(u, v) is a vector function, show how to compute div(g) in this new coordinate system.
If g(u, v) is a vector function, show how to compute curl(g) in this new coordinate system.
If f (u, v) is a scalar function, show how to compute the Laplacian of f in this new coordinate system.
x=
(a)
(b)
(c)
(d)
(e)
3. In class, we discussed 2D irrotational flows, described by scalar and vector potentials, φ and ψ. We showed that
both potentials satisfy Laplace’s equation. There is a much easier way to do this.
(a) If the flow is two dimensional, and u = grad(φ) and u = curl(ψk), then show that the following equations hold.
φy = −ψx .
φx = ψy ,
(b) Supposing that φ and ψ are both twice differentiable, use the result from the previous part to show ∆φ = 0
and ∆ψ = 0.
4. Consider the vector field u = hax + by, cx + dyi, where {a, b, c, d} is a set of nonzero constants. Give conditions on
the constants so that u could be the velocity field of a potential flow. Under these conditions, show that the system
dx
= ax + by
dt
dy
= cx + dy
dt
has a saddle point at the origin, then sketch the vector field and some trajectories. Why is it not surprising that this
equilibrium point isn’t a node, spiral, or center? (Hints: Recall things you learned in DE2. Talk about divergence
and curl.)
1