A metal disc of radius a rotates with a constant angular velocity `omega` about its axis. The potential difference between the center and the rim of the disc is `("m = mass of electron, e = charge on electro")`

Options

A. `(momega^(2)a^(2))/(e)`
B. `(1)/(2)(momega^(2)a^(2))/(e)`
C. `(momega^(2)a^(2))/(2m)`
D. `(momega^(2)a^(2))/(m)`

Answer

To find the potential difference between the center and the rim of a rotating metal disc, we can follow these steps:

### Step 1: Understand the System
We have a metal disc of radius \( a \) rotating with a constant angular velocity \( \omega \). We need to find the potential difference between the center of the disc and a point on its rim.

### Step 2: Identify Forces Acting on an Electron
An electron at a distance \( r \) from the center of the disc experiences a centripetal force due to its circular motion. This centripetal force is provided by the electric force acting on the electron due to the electric field \( E \) generated by the rotation of the disc.

### Step 3: Write the Expression for Centripetal Force
The centripetal force \( F_c \) acting on the electron can be expressed as:
\[ F_c = m \omega^2 r \]
where \( m \) is the mass of the electron, \( \omega \) is the angular velocity, and \( r \) is the distance from the center.

### Step 4: Relate Electric Force to Centripetal Force
The electric force \( F_e \) acting on the electron is given by:
\[ F_e = eE \]
where \( e \) is the charge of the electron and \( E \) is the electric field at distance \( r \). Since the electric force provides the centripetal force, we have:
\[ eE = m \omega^2 r \]

### Step 5: Find the Electric Field
From the above equation, we can express the electric field \( E \) as:
\[ E = \frac{m \omega^2 r}{e} \]

### Step 6: Calculate the Potential Difference
The small potential difference \( dV \) when moving a small distance \( dr \) is given by:
\[ dV = E \, dr \]
Substituting the expression for \( E \):
\[ dV = \frac{m \omega^2 r}{e} \, dr \]

### Step 7: Integrate to Find Total Potential Difference
To find the total potential difference \( V \) between the center (where \( r = 0 \)) and the rim (where \( r = a \)), we integrate:
\[ V = \int_0^a \frac{m \omega^2 r}{e} \, dr \]
Calculating the integral:
\[ V = \frac{m \omega^2}{e} \int_0^a r \, dr = \frac{m \omega^2}{e} \left[ \frac{r^2}{2} \right]_0^a = \frac{m \omega^2}{e} \cdot \frac{a^2}{2} \]
Thus, the potential difference \( V \) is:
\[ V = \frac{m \omega^2 a^2}{2e} \]

### Final Answer
The potential difference between the center and the rim of the disc is:
\[ V = \frac{m \omega^2 a^2}{2e} \]

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