Two point charges of `3.2 xx 10^-19 C and -3.2 xx 10^-19 C` are separated from each other by `2.4 xx 10^-10 m`. The dipole is situated in a uniform electric field of intensity `4 xx 10^5 Vm^-1`. Calculate the work done in roating the dipole by `180^(@)`.

Options

A. `61.44 xx 10^-24 J`
B. `61.44 xx 10^-20 J`
C. `41.66 xx 10^-24 J`
D. `41.66 xx 10^-20 J`

Explanation

To calculate the work done in rotating a dipole in a uniform electric field, we can use the formula:

\[ W = -\vec{p} \cdot \vec{E} \cdot (1 - \cos \theta) \]

Where:
- \( W \) is the work done,
- \( \vec{p} \) is the dipole moment,
- \( \vec{E} \) is the electric field intensity,
- \( \theta \) is the angle of rotation.

### Step 1: Calculate the dipole moment \( \vec{p} \)

The dipole moment \( \vec{p} \) is given by:

\[ \vec{p} = q \cdot d \]

Where:
- \( q = 3.2 \times 10^{-19} \, C \) (magnitude of one of the charges),
- \( d = 2.4 \times 10^{-10} \, m \) (distance between the charges).

Substituting the values:

\[ \vec{p} = (3.2 \times 10^{-19} \, C) \cdot (2.4 \times 10^{-10} \, m) \]

Calculating this gives:

\[ \vec{p} = 7.68 \times 10^{-29} \, C \cdot m \]

### Step 2: Use the angle \( \theta \)

In this case, we are rotating the dipole by \( 180^\circ \). Therefore, \( \cos(180^\circ) = -1 \).

### Step 3: Substitute values into the work done formula

Now substituting the values into the work done formula:

\[ W = -\vec{p} \cdot \vec{E} \cdot (1 - \cos(180^\circ)) \]

This simplifies to:

\[ W = -\vec{p} \cdot \vec{E} \cdot (1 - (-1)) \]
\[ W = -\vec{p} \cdot \vec{E} \cdot 2 \]

Now substituting the values of \( \vec{p} \) and \( \vec{E} \):

\[ W = - (7.68 \times 10^{-29} \, C \cdot m) \cdot (4 \times 10^{5} \, V/m) \cdot 2 \]

Calculating this gives:

\[ W = - (7.68 \times 10^{-29} \cdot 4 \times 10^{5} \cdot 2) \]
\[ W = - (7.68 \times 8 \times 10^{-24}) \]
\[ W = - (61.44 \times 10^{-24}) \]
\[ W = - 6.144 \times 10^{-23} \, J \]

### Step 4: Final result

The work done in rotating the dipole by \( 180^\circ \) is:

\[ W = 6.144 \times 10^{-23} \, J \]