Maxwell's Equations - State what arbitary fields describe.

[bmarson123]

Homework Statement

State whether the following arbitary fields can describe either a magnetic field, a magnetostatic field, neither, or both. In each case justify your answer:

i) R(r) = R₀ (x²,y²,z²)

ii) S(r) = S₀ (x, -z, y)

iii) T(r) = T₀ (-z, 0, x)

Homework Equations

div B= 0

The Attempt at a Solution

i) div R = R₀ (2x, 2y, 2z)

Not a magnetic field as div B $\neq$ 0

ii) div S = S₀ (1,0,0)

Not a magnetic field as div B $\neq$ 0

iii) div T = T₀ (0,0,0)

Can represent a magnetic field as div B = 0


I don't know if the above is correct for the magnetic field and I don't have a clue where to start to establish a decision about them being a magnetostatic field or not!

 

[LiorE]

Hint: A magnetostatic field is a solution to one of the Maxwell equations in the case where everything is not dependent on time. Which equation is it?

Also, obviously, in order for something to be magnetostatic, it has to first be a magnetic field.

Your arguments about determining which fields can be magnetic fields are correct.

 

[bmarson123]

curl H = j + d D / dt ??

So is H the magnetostatic field?? And how can I apply that to this? Do I have to do curl of each field?

 

[LiorE]

Well, if something is not a magnetic field, it's not a magnetostatic field either. And yes, that's the correct equation. In the magnetostatic case there are no time-varying fields, so you just need to find j.

 

[bmarson123]

So for cases i) and ii) I can just say they are neither magnetic or magnetostatic, as it can't be magnetostatic if it isn't magnetic?

Which means I only need to find the current density for iii)? I'm pretty sure I don't know how to do that, do I use curl B = $\mu$ j ?

 

[LiorE]

Yep, that's right.

 

[bmarson123]

But what do I say after I've calculated the current density?

I've got curl B = 2 j (j as in vector j)

But what does that tell me anyway?

 

[LiorE]

If I didn't do it wrong, I think it's supposed to be -2j.

What this means is that this field was created by a current density which is uniform all throughout space its direction is the y-axis (or the -y) and its magnitude is 2T_0.

Obviously this kind of current density doesn't exist in real life, but its a perfectly valid source for a magnetic field as far as Maxwell's equations are concerned.