An electric dipole made up of a positive and negative charge, each of `1mu C` separated by a distance of 2cm is placed in an electric field of `10^(5)N//C`, then the work done in rotating the dipole from the position of stable equilibrium through an angle of `180^(@)` is

Options

A. `2 xx 10^(-3)` Joule
B. `2 xx 10^(-8)` Joule
C. `4 xx 10^(-3)` Joule
D. zero

Answer

To solve the problem of calculating the work done in rotating an electric dipole from a position of stable equilibrium through an angle of 180 degrees, we can follow these steps:

### Step 1: Understand the dipole moment
The dipole moment \( p \) is given by the formula:
\[ p = q \cdot l \]
where \( q \) is the charge and \( l \) is the separation distance between the charges.

### Step 2: Substitute the values for the dipole moment
In this case, the charge \( q = 1 \mu C = 1 \times 10^{-6} C \) and the distance \( l = 2 cm = 2 \times 10^{-2} m \). Thus, we can calculate \( p \):
\[ p = (1 \times 10^{-6} C) \cdot (2 \times 10^{-2} m) = 2 \times 10^{-8} C \cdot m \]

### Step 3: Understand the work done in rotating the dipole
The work done \( W \) in rotating the dipole in an electric field \( E \) can be calculated using the formula:
\[ W = -\Delta U = - (U_2 - U_1) \]
where \( U \) is the potential energy given by:
\[ U = -p \cdot E \cdot \cos(\theta) \]
Here, \( \theta_1 = 0^\circ \) (stable equilibrium) and \( \theta_2 = 180^\circ \).

### Step 4: Calculate the potential energy at both angles
1. For \( \theta_1 = 0^\circ \):
\[ U_1 = -p \cdot E \cdot \cos(0) = -p \cdot E \]
2. For \( \theta_2 = 180^\circ \):
\[ U_2 = -p \cdot E \cdot \cos(180) = p \cdot E \]

### Step 5: Substitute the values into the work done formula
Now substituting \( U_1 \) and \( U_2 \):
\[ W = - (p \cdot E - (-p \cdot E)) = - (p \cdot E + p \cdot E) = -2p \cdot E \]

### Step 6: Substitute the values of \( p \) and \( E \)
Given \( E = 10^5 N/C \):
\[ W = -2 \cdot (2 \times 10^{-8} C \cdot m) \cdot (10^5 N/C) \]
\[ W = -4 \times 10^{-3} J \]

### Step 7: Finalize the answer
Since work done is typically expressed as a positive value when considering the magnitude:
\[ W = 4 \times 10^{-3} J \]

Thus, the work done in rotating the dipole from the position of stable equilibrium through an angle of \( 180^\circ \) is \( 4 \times 10^{-3} J \).