Question

An electric dipole with dipole moment 4 x 10^-9 Cm is aligned at 30 degree with the direction of the uniform electric field of magnitude 5 x 10⁴ NC^-1. The magnitude of the torque acting on the diapole is 

• A. 10^-5 Nm
• B. 10^-4 Nm
• C. 10 x 10^-3 Nm
• D. √ 3 x 10^-4 Nm

Answer 1

The Correct Option is B

 The torque acting on an electric dipole in a uniform electric field is given by the formula:

\( \tau = pE \sin \theta \)

Given:

• Dipole moment (p): \( 4 \times 10^{-9} \, \text{Cm} \)
• Electric field (E): \( 5 \times 10^4 \, \text{N/C} \)
• Angle (θ): \( 30^\circ \)

Step 1: Substitute the given values into the formula:

\( \tau = (4 \times 10^{-9} \, \text{Cm})(5 \times 10^4 \, \text{N/C}) \sin(30^\circ) \)

Step 2: Use the value of \( \sin(30^\circ) = \frac{1}{2} \):

\( \tau = (4 \times 10^{-9} \, \text{Cm})(5 \times 10^4 \, \text{N/C}) \times \frac{1}{2} \)

Step 3: Simplify the expression:

\( \tau = (20 \times 10^{-5} \, \text{Cm}) \times \frac{1}{2} = 10 \times 10^{-5} \, \text{Nm} \)

Therefore, the magnitude of the torque acting on the dipole is: \( 10^{-4} \, \text{Nm} \). The correct answer is (B) \( 10^{-4} \, \text{Nm} \).

Answer 2

We are given the following information: 

• Magnitude of the electric dipole moment, \( p = 4 \times 10^{-9} \, \text{Cm} \)
• Magnitude of the uniform electric field, \( E = 5 \times 10^4 \, \text{NC}^{-1} \)
• Angle between the dipole moment vector and the electric field vector, \( \theta = 30^\circ \)

The torque \( \vec{\tau} \) experienced by an electric dipole with dipole moment \( \vec{p} \) when placed in a uniform electric field \( \vec{E} \) is given by the cross product:

\[ \vec{\tau} = \vec{p} \times \vec{E} \]

The magnitude of this torque is given by:

\[ \tau = |\vec{p}| |\vec{E}| \sin\theta \]

Substituting the given values into the formula:

\[ \tau = (4 \times 10^{-9} \, \text{Cm}) \times (5 \times 10^4 \, \text{NC}^{-1}) \times \sin(30^\circ) \]

First, calculate the product of the magnitudes of the dipole moment and the electric field:

\[ p E = (4 \times 10^{-9}) \times (5 \times 10^4) = (4 \times 5) \times (10^{-9} \times 10^4) \]

\[ p E = 20 \times 10^{-9+4} = 20 \times 10^{-5} \, \text{Nm} \]

Now, find the value of \( \sin(30^\circ) \):

\[ \sin(30^\circ) = \frac{1}{2} = 0.5 \]

Finally, calculate the magnitude of the torque:

\[ \tau = (20 \times 10^{-5}) \times \sin(30^\circ) \]

\[ \tau = (20 \times 10^{-5}) \times (0.5) \]

\[ \tau = 10 \times 10^{-5} \, \text{Nm} \]

\[ \tau = 1 \times 10^1 \times 10^{-5} \, \text{Nm} \]

\[ \tau = 1 \times 10^{1-5} \, \text{Nm} \]

\[ \tau = 1 \times 10^{-4} \, \text{Nm} \]

Therefore, the magnitude of the torque acting on the dipole is \( 10^{-4} \, \text{Nm} \).