The Induced Voltage in a Straight Wire with Time Varying Magnetic Field

[mikec2003]

1. I have what I thought was a simple question. But it turns to be more much difficult.

The situation statement:

Suppose I have a straight wire of length H in the plane of the page. Now I apply a time varying magnetic field (dB/dt), where B is into (perpendicular) to the page and spatially invariant. What is the induced voltage across the ends of the wire?Here's what it looks like:
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dB/dt into page | Vemf = ____
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2. Homework Equations E = vxB Vemf = -d(AB)/dt
3. The Attempt at a Solution : I know that a wire moving through a magnetic field will induce an E field = vxB, and v = dx/dt. It seems logical that it would be the same as a stationary wire in a time varying magnetic field. However, I know that:

dB/dt = -curl(E) and from stokes theorem Vemf = -integral(curl(E)*dS) = -integral(curl(E).dl) and these require a closed path to be integrated over.

There must be a solution since i know that a wire in a time varying magnetic field develops a voltage ...does anybody have any ideas?I attempted to change the equation so

 

[hikaru1221]

Hi mikec2003,

where B is [...] spatially invariant.

This is simply impossible. It would require infinite energy to build up such field. Besides, it's also impossible to visualize an induced E-field incorporating with this B-field, right? And one more point: If you try to visualize it, with some reasonable deductions, you will see that it will lead to another paradox, beside the infinite energy one