Electromagnetic waves, Maxwell's Equations, Laplace?

[tjkubo]

Homework Statement

Suppose Maxwell's displacement current was left out of the Maxwell equations. Show that , in a vacuum, the magnetic field has to have the form B = grad f(r,t), where f is any function which satisfies the Laplace equation.

Homework Equations

curl E = - dB/dt
curl B = 0
div E = 0
div B = 0

The Attempt at a Solution

The question requires us to use Maxwell's Equations, however, we're unsure which is the correct starting point. We've already looked very closely at both Gauss's Laws and the Maxwell-Faraday, but are unsure how to derive B = grad f(r,t) from these where f satisfies the Laplace Eqn.

All we know is that if we plug in B = grad f(r,t) into div B = 0, it works, but is that the most general form?
If the curl of a field = 0, doesn't that imply a conservative vector field? Meaning the field has a potential function? Is THAT correct?

 

[stevenb]

The Attempt at a Solution

The question requires us to use Maxwell's Equations, however, we're unsure which is the correct starting point. We've already looked very closely at both Gauss's Laws and the Maxwell-Faraday, but are unsure how to derive B = grad f(r,t) from these where f satisfies the Laplace Eqn.

All we know is that if we plug in B = grad f(r,t) into div B = 0, it works, but is that the most general form?
If the curl of a field = 0, doesn't that imply a conservative vector field? Meaning the field has a potential function? Is THAT correct?

There is the vector identity that says that the curl of the gradient is zero. This is the starting point. From this it follows that B=grad F, and then substituting that into Gauss's Law gives you the Laplace equation.