A parallel plate capacitor of capacitance `10 muF` is connected across a battery of emf 5 mV. Now, the space between the plates of the capacitors is filled with a deielectric material of dielectric constant K=5. Then, the charge that will flow through the battery till equilibrium is reached is

Options

A. `250 muC`
B. `250 nC`
C. `200 nC`
D. `200 muC`

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the initial charge stored in the capacitor
The initial charge \( Q \) stored in a capacitor is given by the formula:
\[ Q = C \times V \]
where:
- \( C \) is the capacitance,
- \( V \) is the voltage (emf of the battery).

Given:
- \( C = 10 \, \mu F = 10 \times 10^{-6} \, F \)
- \( V = 5 \, mV = 5 \times 10^{-3} \, V \)

Substituting the values:
\[ Q = (10 \times 10^{-6}) \times (5 \times 10^{-3}) = 50 \times 10^{-9} \, C = 50 \, nC \]

### Step 2: Calculate the new capacitance after inserting the dielectric
When a dielectric material is inserted between the plates of a capacitor, the new capacitance \( C' \) can be calculated using the formula:
\[ C' = K \times C \]
where \( K \) is the dielectric constant.

Given:
- \( K = 5 \)

Substituting the values:
\[ C' = 5 \times (10 \times 10^{-6}) = 50 \times 10^{-6} \, F = 50 \, \mu F \]

### Step 3: Calculate the new charge stored in the capacitor with the dielectric
The new charge \( Q' \) stored in the capacitor after the dielectric is inserted is given by:
\[ Q' = C' \times V \]

Substituting the values:
\[ Q' = (50 \times 10^{-6}) \times (5 \times 10^{-3}) = 250 \times 10^{-9} \, C = 250 \, nC \]

### Step 4: Calculate the charge that flows through the battery
The charge that flows through the battery until equilibrium is reached is the difference between the new charge and the initial charge:
\[ \Delta Q = Q' - Q \]

Substituting the values:
\[ \Delta Q = 250 \, nC - 50 \, nC = 200 \, nC \]

### Final Answer
The charge that will flow through the battery till equilibrium is reached is:
\[ \Delta Q = 200 \, nC \]

---