Two identical rings `P and Q` of radius `0.1 m` are mounted coaxially at a distance `0.5 m` apart. The charges on the two rings are `2 mu C and 4 mu C`, respectively. The work done in transferring a charge of `5 mu C` from the center of P to that of Q is.

Options

A. 1.28 J
B. 0.72 J
C. 0.144 J
D. 2.24 J

Explanation

To solve the problem step by step, we will calculate the electric potential at the centers of the rings P and Q and then find the work done in transferring a charge from the center of ring P to the center of ring Q.

### Step 1: Calculate the potential at the center of ring P

The potential \( V_P \) at the center of ring P due to its own charge can be calculated using the formula:
\[ V_P = \frac{k \cdot Q_P}{R_P} \]
where:
- \( k = \frac{1}{4\pi\epsilon_0} \approx 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)
- \( Q_P = 2 \, \mu C = 2 \times 10^{-6} \, C \)
- \( R_P = 0.1 \, m \)

Substituting the values:
\[ V_P = \frac{9 \times 10^9 \cdot 2 \times 10^{-6}}{0.1} = 180000 \, V = 1.8 \times 10^5 \, V \]

### Step 2: Calculate the potential at the center of ring Q

The potential \( V_Q \) at the center of ring Q due to its own charge is:
\[ V_Q = \frac{k \cdot Q_Q}{R_Q} \]
where:
- \( Q_Q = 4 \, \mu C = 4 \times 10^{-6} \, C \)
- \( R_Q = 0.1 \, m \)

Substituting the values:
\[ V_Q = \frac{9 \times 10^9 \cdot 4 \times 10^{-6}}{0.1} = 360000 \, V = 3.6 \times 10^5 \, V \]

### Step 3: Calculate the distance from the center of ring P to the center of ring Q

The distance \( d \) between the centers of the two rings is given as \( 0.5 \, m \). The distance from the center of ring P to the center of ring Q can be calculated using the Pythagorean theorem:
\[ d = \sqrt{(0.5)^2 + (0.1)^2} = \sqrt{0.25 + 0.01} = \sqrt{0.26} \approx 0.51 \, m \]

### Step 4: Calculate the potential at the center of ring P due to ring Q

The potential at the center of ring P due to the charge on ring Q is:
\[ V_{PQ} = \frac{k \cdot Q_Q}{d} \]
Substituting the values:
\[ V_{PQ} = \frac{9 \times 10^9 \cdot 4 \times 10^{-6}}{0.51} \approx 705882.35 \, V \]

### Step 5: Calculate the potential at the center of ring Q due to ring P

The potential at the center of ring Q due to the charge on ring P is:
\[ V_{QP} = \frac{k \cdot Q_P}{d} \]
Substituting the values:
\[ V_{QP} = \frac{9 \times 10^9 \cdot 2 \times 10^{-6}}{0.51} \approx 352941.18 \, V \]

### Step 6: Calculate the total potential at the centers

The total potential at the center of ring P:
\[ V_{totalP} = V_P + V_{PQ} = 1.8 \times 10^5 + 705882.35 \approx 8.5888235 \times 10^5 \, V \]

The total potential at the center of ring Q:
\[ V_{totalQ} = V_Q + V_{QP} = 3.6 \times 10^5 + 352941.18 \approx 7.941176 \times 10^5 \, V \]

### Step 7: Calculate the work done in transferring the charge

The work done \( W \) in transferring a charge \( q \) from point P to point Q is given by:
\[ W = q \cdot (V_{totalQ} - V_{totalP}) \]
where:
- \( q = 5 \, \mu C = 5 \times 10^{-6} \, C \)

Substituting the values:
\[ W = 5 \times 10^{-6} \cdot (7.941176 \times 10^5 - 8.5888235 \times 10^5) \]
Calculating the difference:
\[ W = 5 \times 10^{-6} \cdot (-647647.35) \approx -3.238 \times 10^{-6} \, J \]

The negative sign indicates that work is done against the electric field.

### Final Answer
The work done in transferring the charge from the center of ring P to that of ring Q is approximately \( -3.238 \times 10^{-6} \, J \).

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