Question

Two electric dipoles of dipole moments \(3.9 \times 10^{-30}\,\text{C m}\) and \(5.2 \times 10^{-30}\,\text{C m}\) are placed in two different uniform electric fields of strengths \(16 \times 10^{4}\,\text{N C}^{-1}\) and \(4 \times 10^{4}\,\text{N C}^{-1}\) respectively. What is the ratio of maximum torque experienced by the electric dipoles?

• A. \(12:1\)
• B. \(9:1\)
• C. \(1:9\)
• D. \(3:1\)

Hint:

Maximum torque on an electric dipole is \(\tau_{\max} = pE\). For ratios, multiply dipole moment and electric field strength for each case.

Answer 1

The Correct Option is D

Step 1: Use torque formula for an electric dipole.
The torque experienced by an electric dipole in a uniform electric field is:
\[ \tau = pE\sin\theta \]

Step 2: Use condition for maximum torque.
Maximum torque occurs when:
\[ \sin\theta = 1 \]
So,
\[ \tau_{\max} = pE \]

Step 3: Write the ratio of maximum torques.
\[ \frac{\tau_1}{\tau_2} = \frac{p_1E_1}{p_2E_2} \]

Step 4: Substitute given values.
\[ p_1 = 3.9 \times 10^{-30}, \quad E_1 = 16 \times 10^4 \]
\[ p_2 = 5.2 \times 10^{-30}, \quad E_2 = 4 \times 10^4 \]
\[ \frac{\tau_1}{\tau_2} = \frac{3.9 \times 10^{-30} \times 16 \times 10^4} {5.2 \times 10^{-30} \times 4 \times 10^4} \]

Step 5: Cancel common powers.
\[ \frac{\tau_1}{\tau_2} = \frac{3.9 \times 16}{5.2 \times 4} \]

Step 6: Simplify.
\[ \frac{\tau_1}{\tau_2} = \frac{62.4}{20.8} \]
\[ \frac{\tau_1}{\tau_2} = 3 \]

Step 7: Final answer.
\[ \boxed{3:1} \]