Finding the B field in a long cylindrical hole in a long wire

AI Thread Summary
The discussion revolves around calculating the magnetic field inside a cylindrical cavity within a cylindrical wire carrying a uniform electric current. The user initially attempts to find the vector potential by integrating over the entire cylinder and then subtracting the contribution from the cavity, leading to complex elliptic integrals. However, participants suggest that a simpler approach using Ampere's law would suffice, eliminating the need for the complicated vector potential calculations. The user acknowledges this feedback and realizes that their initial method was unnecessarily complicated. This highlights the importance of exploring simpler methods in electromagnetic problems.
VortexLattice
Messages
140
Reaction score
0
So, I'm trying this problem and just want to make sure my attack is correct. We have an infinitely long cylindrical hole in an infinitely long cylindrical wire, like in this picture:

11bca21b-9df4-47b5-8ce5-d451a1e43490.png


(Here, they call the radius of the hole 'a'. I'm going to call it r.)

There's an electric current I in the wire and the current density is uniform. I want to find the B field anywhere in the cavity. First, I find J, the current density:

J = \frac{I}{\pi (R^2 - r^2)}

So, the step I'm most worried about is this: I want to find the vector potential A in the hole, but it's pretty hard to integrate around a hole. So I'm going to find the vector potential due to the whole cylinder (as if there were no hole), and then the vector potential due to a current of the same density going through the hole, and then subtract the latter from the former (because it's just a superposition, like electric potential, right?).

I'm doing this in polar coordinates, where the z axis goes along the center of the wire. So, typically you have to integrate over a volume to get A, but we don't actually care about the z direction here, so we just integrate it from 0 to z, and it should cancel out at the end:

\vec{A}(\vec{x}) = Jz \int_0^R r'dr' \int_0^{2\pi} d\phi' \frac{1}{\left|\vec{x} - \vec{x'}\right|}

So, I say that our observation point \vec{x} is (x,0,z), and the point \vec{x'} we're integrating at is (x',y',z). Then, in polar coordinates, we have \vec{x'} = (r'cos(\phi'),r'sin(\phi'),z). Plugging this in, I get:


\vec{A}(\vec{x}) = Jz \int_0^R r'dr' \int_0^{2\pi} d\phi' \frac{1}{\sqrt{(x - r'cos(\phi'))^2 + (r'sin(\phi'))^2}} = Jz \int_0^R r'dr' \int_0^{2\pi} d\phi' \frac{1}{\sqrt{x^2 -2xr'cos(\phi') + r'^2}}

Now, I integrate. I actually first integrate over r'. I do a kind of completing the square thing in the denominator, and then integrate:

\vec{A}(\vec{x}) = Jz \int_0^R r'dr' \int_0^{2\pi} d\phi' \frac{1}{\sqrt{(r' - cos(\phi'))^2 + x^2sin^2(\phi')}}

I integrated this in wolfram, and it simplifies a little. But now I have to integrate over phi, and it gives me an elliptic integral of the first kind. Academia has ruined me so I get afraid when an answer doesn't boil down to three terms or less, so I'm just wondering if this seems right or not.

Also, is my general idea a good one or is there a much easier way?

Thanks!
 
Physics news on Phys.org
So, it has been pointed out to me that I am functionally retarded: You use the premise I'm using (subtracting the current due to a cylinder in place of the hole), but you don't do the whole silliness with the vector potential. Because you're calculating them separately, you can just use Ampere's law. Yayyy, I'm stupid.
 
Thread 'Gauss' law seems to imply instantaneous electric field'
Imagine a charged sphere at the origin connected through an open switch to a vertical grounded wire. We wish to find an expression for the horizontal component of the electric field at a distance ##\mathbf{r}## from the sphere as it discharges. By using the Lorenz gauge condition: $$\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t}=0\tag{1}$$ we find the following retarded solutions to the Maxwell equations If we assume that...
I passed a motorcycle on the highway going the opposite direction. I know I was doing 125/km/h. I estimated that the frequency of his motor dropped by an entire octave, so that's a doubling of the wavelength. My intuition is telling me that's extremely unlikely. I can't actually calculate how fast he was going with just that information, can I? It seems to me, I have to know the absolute frequency of one of those tones, either shifted up or down or unshifted, yes? I tried to mimic the...

Similar threads

Back
Top