Question

A conducting rod of length l is hinged at point O. It is free to rotate in a vertical plane. There exists
a uniform magnetic field B (into the plane)

in horizontal direction. The rod is released from the position shown. The
potential difference between the two ends of the rod is proportional to

Answer 1

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Answer to Question #70009, Physics / Electromagnetism

Question:

A conducting rod of length $l$ is hinged at point O. It is free to rotate in a vertical plane. There exists a uniform magnetic field B (into the plane) in horizontal direction. The rod is released from the position shown. The potential difference between the two ends of the rod is proportional to

Solution:

In general the potential difference between two points on the conductor moving in the magnetic field can be calculated as

$$U = - \frac {d \Phi}{d t}$$

Where $\frac{d\Phi}{dt}$ is the rate of change of the magnetic flux

In this task

$$\frac {d \Phi}{d t} = \frac {d (S B)}{d t} = B \frac {d \left(\frac {l ^ {2}}{2} \varphi\right)}{d t} = \frac {l ^ {2} B}{2} \left(\frac {d \varphi}{d t}\right)$$

So the potential difference between the two ends of the rod is proportional to

$$l ^ {2} \text{ and } B$$