A conducting rod PQ is rotated in a magnetic field B about an axis passing through point O as shown in figure. Then potential difference between P&Q is (`omega` : angular speed)

Options

A. `[B omega (b^2-a^2)]`
B. `(B omega b^2) /2`
C. `(B omega a^2) /2`
D. `[(B omega)/2 (b^2-a^2)]`

Answer

To find the potential difference between points P and Q in a conducting rod PQ that is rotating in a magnetic field B about an axis through point O, we can follow these steps:

### Step 1: Understand the setup
The conducting rod PQ is rotating in a magnetic field. The rotation creates a change in magnetic flux through the area swept by the rod, which induces an electromotive force (emf) between the ends of the rod.

### Step 2: Determine the area swept by the rod
The area swept out by the rod when it rotates can be expressed as:
\[ A = \pi (b^2 - a^2) \]
where \( b \) is the distance from the axis of rotation to point Q, and \( a \) is the distance from the axis of rotation to point P.

### Step 3: Calculate the magnetic flux
The magnetic flux \( \Phi \) through the area swept by the rod is given by:
\[ \Phi = B \cdot A = B \cdot \pi (b^2 - a^2) \]
where \( B \) is the magnetic field strength.

### Step 4: Relate time to angular velocity
The time \( t \) taken for one complete rotation can be expressed in terms of angular velocity \( \omega \):
\[ t = \frac{2\pi}{\omega} \]

### Step 5: Calculate the induced emf
The induced emf \( E \) is given by the rate of change of magnetic flux:
\[ E = -\frac{d\Phi}{dt} \]
Substituting the expression for magnetic flux and the time period, we have:
\[ E = -\frac{B \cdot \pi (b^2 - a^2)}{t} = -\frac{B \cdot \pi (b^2 - a^2)}{\frac{2\pi}{\omega}} \]
This simplifies to:
\[ E = \frac{B \cdot \omega (b^2 - a^2)}{2} \]

### Step 6: Final expression for potential difference
Thus, the potential difference \( V \) between points P and Q is:
\[ V = E = \frac{B \cdot \omega (b^2 - a^2)}{2} \]

### Conclusion
The potential difference between points P and Q in the rotating conducting rod is:
\[ V = \frac{B \cdot \omega (b^2 - a^2)}{2} \]

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