Induced Current

[nahya]

This problem is difficult to describe, so I'll post a picture.
http://img71.imageshack.us/my.php?image=pic1ik.gif

The figure above shows a rod of length L caused to move at a constant speed v along horizontal conducting rails. The magnetic field B (the magnitude and direction of which are qualitatively shown by the figure) is not constant, but is supplied by a long wire parallel to the conducting rails. This wire is a distance a from the rail and has a current i.

L=3.13 cm, v=3.11 m/s, a=15.6 mm, and i=11 A.

What is the induced emf (e) in the rod?

---
B = (u_0 I)/(2pi y), and it is not uniform, so I integrated over y=a...L
I got (u_0 I)/(2pi)*ln(L/a).

emf = vBL = v * (u_0 I)/(2pi)*ln(L/a) * L = 3.11 * 1.544597242E-6 * 0.0313 = 1.503557293E-7 V

That is not the right answer, however.
I double-checked that my calculations are correct. So I'm guessing that my steps are incorrect. Can anyone point me to where I'm going wrong?

Thanks.

[Hootenanny]

I would try this;

B = \frac{\mu_{0} I}{2\pi y}

a is a constant; y = a + L

\frac{dB}{dL} = \frac{\mu_{0} I}{2\pi a + 2\pi L} \; dL

B = \int^{0.0313}_{0} \frac{\mu_{0} I}{2\pi a + 2\pi L} \; dL

See if that works.

~H

[Päällikkö]

Shouldn't the integration be done over y=a..a+L?

EDIT: I'm always too slow :smile:.

[nahya]

B was approximately 3.326E-4, and emf was 3.23762818E-5 V, which was still incorrect.
Am I using the right L for emf = vBL? I'm using 3.13 cm (0.0313 m).

[Hootenanny]

I think you may have integrated in correctly you should obtain;

B = \int^{0.0313}_{0} \frac{\mu_{0} I}{2\pi a + 2\pi L} \; dL

B = \left[ \frac{1}{2}\mu_{0}I \log (a + L) \right]^{0.0313}_{0}

Also ensure that you are converting correctly, note that a is given in mm. You are using the correct equation here;

Am I using the right L for emf = vBL? I'm using 3.13 cm (0.0313 m)

~H

[nahya]

Thanks, that worked.
Weird, though, because I had used my calculator to graph the equation and integrate graphically.

[Hootenanny]

Thanks, that worked.
Weird, though, because I had used my calculator to graph the equation and integrate graphically.

No problem. I prefer pen and paper :wink:

~H