Question

A Combination of two charges +1 nC and -1 nC are separated by a distance of 1 μm. This Constituted electric dipole is placed in an electric field of 1000 V/m at an angle of 45 degree. The torque and the potential energy on the electric dipole are:

• A. \(\frac{1}{\sqrt{2}}\) x10^-12 N.m and \(\frac{1}{\sqrt{2}}\) x10^-12 J
• B. \(\frac{1}{\sqrt{2}}\) x10^-12 N.m and \(\sqrt{2}\) x10^-12 J
• C. \(\sqrt{2}\) x10^-12 N.m and \(\frac{1}{\sqrt{2}}\) x10^-12 J
• D. \(\sqrt{2}\) x10^-12 N.m and \(\sqrt{2}\) x10^-12 J
• E. \(\frac{\sqrt{3}}{2}\) x10^-12 N.m and \(\frac{\sqrt{3}}{2}\)x10^-12 J

Answer 1

The Correct Option is A

Given parameters:

• Charges: \( q = ±1 \, \text{nC} = ±10^{-9} \, \text{C} \)
• Separation: \( d = 1 \, \mu\text{m} = 10^{-6} \, \text{m} \)
• Electric field: \( E = 1000 \, \text{V/m} \)
• Angle: \( θ = 45° \)

 

Dipole moment calculation: \[ p = q \times d = 10^{-9} \times 10^{-6} = 10^{-15} \, \text{C.m} \]

Torque on dipole: \[ \tau = pE \sinθ = 10^{-15} \times 1000 \times \sin45° = \frac{10^{-12}}{\sqrt{2}} \, \text{N.m} \]

Potential energy: \[ U = -pE \cosθ = -10^{-15} \times 1000 \times \cos45° = -\frac{10^{-12}}{\sqrt{2}} \, \text{J} \]

Thus, the correct option is (A): \( \frac{1}{\sqrt{2}} \times 10^{-12} \, \text{N.m} \) and \( \frac{1}{\sqrt{2}} \times 10^{-12} \, \text{J} \).

Answer 2

1. Calculate the dipole moment (p):

The electric dipole moment (p) is given by:

\[p = qd\]

where:

• q = 1 nC = 1 × 10⁻⁹ C (magnitude of charge)
• d = 1 μm = 1 × 10⁻⁶ m (separation between charges)

\[p = (1 \times 10^{-9} \, C)(1 \times 10^{-6} \, m) = 1 \times 10^{-15} \, C \cdot m\]

2. Calculate the torque (τ):

The torque (τ) on an electric dipole in an electric field (E) is given by:

\[\tau = pE\sin\theta\]

where θ = 45° is the angle between the dipole moment and the electric field.

\[\tau = (1 \times 10^{-15} \, C \cdot m)(1000 \, V/m)\sin(45^\circ) = 10^{-12} \frac{1}{\sqrt{2}} \, N \cdot m\]

3. Calculate the potential energy (U):

The potential energy (U) of an electric dipole in an electric field is given by:

\[U = -pE\cos\theta\]

\[U = -(1 \times 10^{-15} \, C \cdot m)(1000 \, V/m)\cos(45^\circ) = -10^{-12} \frac{1}{\sqrt{2}} \, J\]

The question asks for the potential *energy*, so we're interested in the magnitude:

\[|U| = \frac{1}{\sqrt{2}} \times 10^{-12} \, J\]