Question

An electric dipole with dipole moment \( \vec{P} = (2.54 \times 10^{-28} \text{ C.m}) (2.00\hat{i} + 3.00\hat{j}) \) is placed in an electric field \( \vec{E} = \left( 1000 \frac{\text{N}}{\text{C}} \right) \hat{i \). An external agent turns the dipole until its electric dipole moment is \( \vec{P} = (2.54 \times 10^{-28} \text{ C.m}) (- 3.00\hat{i} + 2.00\hat{j}) \). The work done by the agent is :

• A. \( 2.54 \times 10^{-25} \text{ J} \)
• B. \( 1.27 \times 10^{-24} \text{ J} \)
• C. \( 1.02 \times 10^{-23} \text{ J} \)
• D. \( 2.29 \times 10^{-26} \text{ J} \)

Hint:

When dealing with vectors in dipole problems, it is much faster and less error-prone to use the vector dot product directly \( W = \vec{E} \cdot (\vec{P}_i - \vec{P}_f) \) rather than calculating angles and using the \( pE(\cos\theta_1 - \cos\theta_2) \) formula.

Answer 1

The Correct Option is B

Concept:
• When an electric dipole with dipole moment \( \vec{P} \) is placed in a uniform electric field \( \vec{E} \), it possesses electrostatic potential energy.

• This potential energy is given by the dot product: \( U = -\vec{P} \cdot \vec{E} \).

• To slowly rotate a dipole from an initial orientation to a final orientation, an external agent must do work against the electric field.

• The total work done by the external agent equals the change in the dipole's potential energy: \( W_{\text{ext}} = \Delta U = U_{\text{final}} - U_{\text{initial}} \).

Step 1: Express the given vectors and the work energy formula
Let the scalar constant be \( p_0 = 2.54 \times 10^{-28} \text{ C.m} \).
The initial dipole moment is \( \vec{P}_i = p_0 (2.00\hat{i} + 3.00\hat{j}) \).
The final dipole moment is \( \vec{P}_f = p_0 (-3.00\hat{i} + 2.00\hat{j}) \).
The uniform electric field is \( \vec{E} = 1000\hat{i} \text{ N/C} \).
The work done formula is \( W = (-\vec{P}_f \cdot \vec{E}) - (-\vec{P}_i \cdot \vec{E}) = (\vec{P}_i - \vec{P}_f) \cdot \vec{E} \).

Step 2: Perform the vector subtraction and dot product
First, calculate the difference vector \( (\vec{P}_i - \vec{P}_f) \):
\[ \vec{P}_i - \vec{P}_f = p_0 [ (2.00\hat{i} + 3.00\hat{j}) - (-3.00\hat{i} + 2.00\hat{j}) ] \]
\[ \vec{P}_i - \vec{P}_f = p_0 [ (2.00 + 3.00)\hat{i} + (3.00 - 2.00)\hat{j} ] \]
\[ \vec{P}_i - \vec{P}_f = p_0 (5.00\hat{i} + 1.00\hat{j}) \]
Next, perform the dot product with the electric field \( \vec{E} = 1000\hat{i} \):
\[ W = [ p_0 (5.00\hat{i} + 1.00\hat{j}) ] \cdot [ 1000\hat{i} ] \]
Since \( \hat{i} \cdot \hat{i} = 1 \) and \( \hat{j} \cdot \hat{i} = 0 \), only the x-component survives.
\[ W = p_0 \times 5.00 \times 1000 \]
\[ W = 5000 \times (2.54 \times 10^{-28}) \text{ J} \]

Step 3: Calculate final numerical value
\[ W = 12700 \times 10^{-28} \text{ J} \]
Rewrite this in proper scientific notation by shifting the decimal 4 places to the left:
\[ W = 1.27 \times 10^4 \times 10^{-28} \text{ J} \]
\[ W = 1.27 \times 10^{-24} \text{ J} \]

Step 4: Conclusion
The total work done by the external agent is mathematically calculated to be \( 1.27 \times 10^{-24} \text{ J} \).
Therefore, option (B) is the correct answer.