(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

See picture for clarity.

There are 2 infinite wires carrying a current i, one wire does it in the upward direction and the other wire in the downward direction. They are separated by a distance L+2d.

Between them lies a circuit such that the part that moves with a velocity v upward has a resistance R and the rest of the circuit has a negligible resistance.

1)Calculate the induced current through the rod moving at constant velocity v.

2)Calculate the force needed to maintain the constant velocity. (applied by an external agent)

3)The work done by the external agent by unit of time.

2. Relevant equations

None given.

3. The attempt at a solution

First I calculated the magnetic field between the 2 wires. I used Ampere's law. I reached [tex]B=\frac{\mu _0 I}{2\pi r}+\frac{\mu _0 I}{2\pi (L+2d-r)}[/tex], pointing into the sheet.

Now [tex]V=RI\Rightarrow I=\frac{V}{R}=-\frac{d\Phi _B}{dt}\cdot \frac{1}{R}[/tex].

So I must find [tex]\frac{d \Phi _B}{dt}[/tex].

First, I'm looking for [tex]\Phi _B[/tex].

I believe that [tex]\Phi _B = A\cdot B_{\text{enclosed}}[/tex].

So I integrated the expression I had found of B from d to L-d, which gave me [tex]B_{\text{enclosed}}=\frac{\mu _0 I}{2\pi r}\left [ \ln \left ( \frac{L-d}{d} \right) + \ln \left ( \frac{d+L}{3d} \right ) \right ][/tex].

Then [tex]\frac{d \Phi _B}{dt}=LvB_{\text{enclosed}}[/tex], thus [tex]I=-\frac{Lv B_{\text{enclosed}}}{R}[/tex].

It was so messy to calculate, I'm sure at 99.9999999999% I've made at least an error...

Could someone confirm it? Or at least tell me if my reasoning was right.

Thanks in advance.

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# Homework Help: Induced emf in a wire hard mathematically

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