Question

An electric dipole of length \(3\text{cm}\) is placed with its axis making an angle of \(60^{\circ}\) with a uniform electric field of \(5\times 10^4\) N/C. It experiences a torque of \(9\sqrt{3}\) Nm. The magnitude of charge on the dipole is \((sin60^{\circ} = \frac{\sqrt{3}}{2})\)

• A. \(0.6\times 10^{-2}\) C
• B. \(1.2\times 10^{-2}\) C
• C. \(1.8\times 10^{-2}\) C
• D. \(2.4\times 10^{-2}\) C

Hint:

Torque \(\tau=pE\sin\theta\) with \(p=q\times2l\) (the dipole length).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
A dipole in a uniform field feels a torque \(\tau=pE\sin\theta\), where \(p=q\times\ell\) and \(\ell\) is the length of the dipole.

Step 2: Key Formula or Approach
\[ \tau=q\,\ell\,E\sin\theta \Rightarrow q=\frac{\tau}{\ell E\sin\theta} \]

Step 3: Detailed Explanation
\(\ell=3\ \text{cm}=0.03\) m, \(E=5\times10^4\) N/C, \(\sin60^{\circ}=\dfrac{\sqrt3}{2}\).
\[ \ell E\sin\theta=0.03\times5\times10^4\times\frac{\sqrt3}{2}=750\sqrt3 \]
\[ q=\frac{9\sqrt3}{750\sqrt3}=0.012=1.2\times10^{-2}\ \text{C} \]

Final Answer:
The charge on the dipole is \(1.2\times10^{-2}\) C, option (B). \[ \boxed{1.2\times10^{-2}\ \text{C (B)}} \]