Question

A dipole of charge 0.01C and separation 0.4mm, is placed in an electric field of strength 10 dyne/CC , Find the maximum torque exerted on the dipole in the field.

• A. 4 x 10^-9 Nm
• B. 2 x 10^-10 Nm
• C. 4 x 10^-10 Nm
• D. 2 x 10^-9 Nm

Answer 1

The Correct Option is C

Step 1: Given parameters

We are given the following values:

• \( P = 0.01 \times 0.4 \times 10^{-3} = 4 \times 10^{-6} \, \text{cm} \)
• \( E = 10 \times 10^{-5} \, \text{N} \)

Step 2: Calculating the Torque

The torque \( \tau \) is calculated as the cross product of the position vector \( \mathbf{P} \) and the electric field \( \mathbf{E} \). The magnitude of the torque is given by:

\[ |\tau| = |\mathbf{P} \times \mathbf{E}| \] 

Step 3: Substituting the values

Substituting the given values of \( \mathbf{P} \) and \( \mathbf{E} \) into the equation:

\[ |\tau| = 4 \times 10^{-6} \times 10 \times 10^{-5} \] 

Step 4: Final calculation

After performing the multiplication, we get:

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

Conclusion:

The magnitude of the torque is \( 4 \times 10^{-10} \, \text{Nm} \)

Answer 2

To find the maximum torque exerted on the dipole in the electric field, we use the formula for the torque on an electric dipole in an electric field:
\(\tau = p \cdot E \cdot \sin \theta\)
Given:
Charge of each dipole \(q = 0.01\)C
Separation between charges d = 0.4mm = \(0.4 \times 10^{-3}\)m,
Electric field strength E = 10dyne/CC \(= 10 \times 10^{-5}\) N/C

Let's, calculate the dipole moment p:
\(p = q \cdot d = 0.01 \text{ C} \cdot 0.4 \times 10^{-3} \text{ m} = 4 \times 10^{-6} \text{ C}\cdot\text{m}\)

Now, calculate the torque \(\tau\):
\(\tau = p \cdot E \cdot \sin \theta\)

To find the maximum torque, we need to consider the maximum value of \(\sin \theta\), which occurs when \(\theta = 90^\circ\) (perpendicular alignment):
\(\sin 90^\circ = 1\)
Therefore,
\(\tau_{\text{max}} = p \cdot E \cdot 1 = p \cdot E\)
\(\tau_{\text{max}} = 4 \times 10^{-6} \text{ C}\cdot\text{m} \cdot 10 \times 10^{-5} \text{ N/C}\)
\(\tau_{\text{max}} = 4 \times 10^{-6} \times 10 \times 10^{-5} \text{ N}\cdot\text{m}\)
\(\tau_{\text{max}} = 4 \times 10^{-6 + 1 - 5} \text{ N}\cdot\text{m}\)
\(\tau_{\text{max}} = 4 \times 10^{-10} \text{ Nm}\)

So, the option is (C): \(4 \times 10^{-10} \text{ Nm}\).