Question

An electric dipole of length \(4\,\mathrm{cm}\) is placed in a uniform electric field of intensity \(2\times10^5\,\mathrm{N\,C^{-1}}\). When the dipole is oriented at a certain angle with the electric field, it experiences a torque of \(0.2\sqrt3\,\mathrm{N\,m}\) and possesses a potential energy of \(-0.2\,\mathrm{J}\). The magnitude of each charge of the dipole is

• A. \(75\,\mu\mathrm{C}\)
• B. \(50\,\mu\mathrm{C}\)
• C. \(100\,\mu\mathrm{C}\)
• D. \(25\,\mu\mathrm{C}\)

Hint:

For an electric dipole, \[ \boxed{ \tau=pE\sin\theta, \qquad U=-pE\cos\theta } \] and \[ \boxed{ p=qd. } \]

Answer 1

The Correct Option is B

Step 1: Write the expressions for torque and potential energy. \[ \tau=pE\sin\theta, \] \[ U=-pE\cos\theta. \] Given, \[ \tau=0.2\sqrt3\,\mathrm{N\,m}, \] \[ U=-0.2\,\mathrm{J}. \]

Step 2: Find the dipole moment. Squaring and adding, \[ (pE)^2 = \tau^2+U^2. \] Thus, \[ pE = \sqrt{(0.2\sqrt3)^2+(0.2)^2} = \sqrt{0.12+0.04} = 0.4. \] Hence, \[ p = \frac{0.4}{2\times10^5} = 2\times10^{-6}\,\mathrm{C\,m}. \]

Step 3: Calculate the charge. Dipole moment is \[ p=qd, \] where \[ d=4\,\mathrm{cm}=0.04\,\mathrm{m}. \] Therefore, \[ q = \frac{2\times10^{-6}}{0.04} = 5\times10^{-5}\,\mathrm{C} = 50\,\mu\mathrm{C}. \] Hence, \[ \boxed{50\,\mu\mathrm{C}} \] Therefore, \[ \boxed{(B)} \] is the correct answer.