Question

An electric dipole is placed at an angle of $30^\circ$ with an electric field intensity $3 \times 10^5 \text{ NC}^{-1}$ experiences a torque $3 \text{ Nm}$. If the length of dipole is $2 \text{ cm}$, the charge on the dipole is (in milli coloumb)}

• A. 2
• B. 3
• C. 1.5
• D. 1
• E. 4

Hint:

Be careful with the length of the dipole. In the formula $p = q(2l)$, the variable '2l' represents the entire separation between the charges, which is given as 2 cm here.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The torque ($\tau$) experienced by a dipole of dipole moment $p$ in an external electric field $E$ at an angle $\theta$ is given by $\tau = pE \sin \theta$.
Key Formula or Approach:
1. Dipole moment $p = q \times (2l)$, where $2l$ is the length of the dipole.
2. Torque $\tau = (q \cdot 2l) E \sin \theta$.

Step 2: Detailed Explanation:
Given:
Torque $\tau = 3\text{ Nm}$
Angle $\theta = 30^\circ$
Electric field $E = 3 \times 10^5 \text{ NC}^{-1}$
Dipole length $2l = 2\text{ cm} = 0.02\text{ m}$
Using the formula:
\[ 3 = q \times 0.02 \times (3 \times 10^5) \times \sin 30^\circ \]
\[ 3 = q \times 0.02 \times 3 \times 10^5 \times 0.5 \]
\[ 3 = q \times 0.01 \times 3 \times 10^5 \]
\[ 3 = q \times 3 \times 10^3 \]
\[ 1 = q \times 10^3 \]
\[ q = 10^{-3} \text{ C} = 1\text{ mC} \]

Step 3: Final Answer:
The charge on the dipole is 1 milli coulomb.