Finding Induced Current Density (J) in a Cylinder under a Magnetic Field

[denislemenoir]

Given a cylinder of length L, radius a and conductivity sigma, how does one find the induced currenty density (J) as a function of p when a magnetic field B is applied?

Where p is the distance from the axis of the cylinder and B is applied along the axis of the cylinder, B = Bosin(wt).

(Neglecting any additional fields due to the induced current)

Hence, how does one calculate the power dissipated in the cylinder?

Thanks
Denis

 

[Andrew Mason]

Given a cylinder of length L, radius a and conductivity sigma, how does one find the induced currenty density (J) as a function of p when a magnetic field B is applied?

I am confused by the setup here. Is it a hollow cylinder? If so, what is its thickness?

I can't help answer your question until I clarify the problem. But consider these questions:

Which law applies here? (Hint it is not Ampere's law or either of Gauss' laws).

What is the flux and the rate of change of flux through the area enclosed by the the cylinder? How is that related to the induced voltage? How is that related to the current and conductivity (and length and area)?

I am a little confused about the current density (J=I/A) in this case because I am not sure what the cross sectional area is.

What direction does the current flow?

AM

 

[denislemenoir]

I am confused by the setup here. Is it a hollow cylinder? If so, what is its thickness?

I can't help answer your question until I clarify the problem. But consider these questions:

Which law applies here? (Hint it is not Ampere's law or either of Gauss' laws).

What is the flux and the rate of change of flux through the area enclosed by the the cylinder? How is that related to the induced voltage? How is that related to the current and conductivity (and length and area)?

I am a little confused about the current density (J=I/A) in this case because I am not sure what the cross sectional area is.

What direction does the current flow?

AM

Hi, the cylinder is not hollow. I tried initially to use Faraday's law of EM induction and all the equations of J I knew, such as...

grad(J) + dD/dt = 0 [where D is the displacement]
and J = -conductivity*grad(potential)

However, I just couldn't get anywhere with them.

I know the solution is J = 0.5*sigma*p*w*Bocoswt [1] [given]

which implies J = 0.5*p*sigma*dB/dt [2]

but I don't understand why equation [2] is true, I've never seen it before. The radius is a, but clearly the cross-sectional area is not relevant here. I'm sure this problem has a very simple explanation, but it escapes me!

Thanks
Denis

 

[clive]

I'm not very convinced that Eq. (2) can really describe the currents inside the cylinder. It seems that currents are very strong at the surface of the cylinder and zero on its axis (because of the proportionality with respect to p).

In my opinion, the problem seems a classical Foucault problem. The variable axial magnetic field induces a (circular/circumferential) electric field. This field will induce circular currents (Foucault) inside the cylinder, currents whose radii depend on the magnetic field amplitude and frequency and are uniformly distributed in the volume of the cylinder. The solution must be independent of the cylinder dimensions unless the cylinder is very small and induces limitations in electrons' motion (like the foils of an AC transformer).