A copper disc of radius 10 cm is rotating in magnetic field B=0.4 gauss withh 10 rev/sec. What will be potential difference across pertipheral pints of disc.

Options

A. `pi uV`
B. `10 pi uV`
C. zero
D. `5 pi u V`

Answer

To find the potential difference across the peripheral points of a rotating copper disc in a magnetic field, we can follow these steps:

### Step 1: Understand the Problem
We have a copper disc of radius \( r = 10 \) cm (or \( 0.1 \) m) rotating in a magnetic field \( B = 0.4 \) gauss (which is \( 0.4 \times 10^{-4} \) T) at a speed of \( 10 \) revolutions per second.

### Step 2: Convert Units
Convert the magnetic field from gauss to tesla:
\[ B = 0.4 \text{ gauss} = 0.4 \times 10^{-4} \text{ T} = 4 \times 10^{-5} \text{ T} \]

### Step 3: Calculate Angular Velocity
The angular velocity \( \omega \) in radians per second is given by:
\[ \omega = 2\pi \times \text{(revolutions per second)} = 2\pi \times 10 \text{ rev/sec} = 20\pi \text{ rad/sec} \]

### Step 4: Induced EMF Calculation
The induced EMF (\( \mathcal{E} \)) across a small element of the disc can be calculated using the formula:
\[ \mathcal{E} = B \cdot L \cdot v \]
where \( L \) is the length of the conductor (which is the small element \( dx \)) and \( v \) is the linear velocity of the small element at distance \( x \) from the center, given by:
\[ v = \omega \cdot x \]
Thus, the induced EMF for a small element \( dx \) at radius \( x \) is:
\[ d\mathcal{E} = B \cdot (dx) \cdot (\omega \cdot x) \]

### Step 5: Integrate to Find Total EMF
To find the total EMF across the entire radius of the disc, we integrate from \( 0 \) to \( r \):
\[ \mathcal{E} = \int_0^r B \cdot \omega \cdot x \, dx \]
Calculating the integral:
\[ \mathcal{E} = B \cdot \omega \cdot \int_0^r x \, dx = B \cdot \omega \cdot \left[ \frac{x^2}{2} \right]_0^r = B \cdot \omega \cdot \frac{r^2}{2} \]

### Step 6: Substitute Values
Substituting the values:
\[ \mathcal{E} = (4 \times 10^{-5} \, \text{T}) \cdot (20\pi \, \text{rad/sec}) \cdot \frac{(0.1 \, \text{m})^2}{2} \]
Calculating:
\[ \mathcal{E} = (4 \times 10^{-5}) \cdot (20\pi) \cdot \frac{0.01}{2} = (4 \times 10^{-5}) \cdot (20\pi) \cdot 0.005 \]
\[ \mathcal{E} = 4 \times 10^{-5} \cdot 20 \cdot \pi \cdot 0.005 = 4 \times 10^{-5} \cdot 0.1\pi = 4 \times 10^{-6} \pi \, \text{V} \]

### Step 7: Evaluate the Potential Difference
However, since the disc is symmetric and all peripheral points are equidistant from the center, the potential difference across any two peripheral points will be zero. Therefore, the final answer is:
\[ \text{Potential Difference} = 0 \, \text{V} \]

### Final Answer
The potential difference across the peripheral points of the disc is \( 0 \, \text{V} \).

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