Electric Dipole

Practice Questions:

Question 1: moderate

An electric dipole of moment ‘p’ is placed in an electric field of intensity ‘E’. The dipole acquires a position such that the axis of the dipole makes an angle θ with the direction of the field. Assuming that the potential energy of the dipole to be zero when θ = 90°. the torque and the potential energy of the dipole will respectively be :

1. pE sin θ, 2pE cos θ

2. pE cos θ, – pE sin θ

3. pE sin θ, –pE cos θ

4. pE sin θ, –2pE cos θ

View Answer

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

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

The potential energy of the dipole is defined as:

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

Here, the potential energy is zero when \( \theta = 90^\circ \), which aligns with the condition provided.

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Question 2: moderate

An electric dipole with dipole moment

\( \overrightarrow{p}=\left( 3\hat{i}+4\hat{j} \right) \) C-m, is kept in electric field \(\overrightarrow{E}=0.4kN/C\hat{i} \). What is the torque acting on it & the potential energy of the dipole ?

1. \[ 1600\left( N\times m \right)\hat{k},-1200J\]

2. \[ -1600\left( N\times m \right)\hat{k},1200J\]

3. \[ -1600\left( N\times m \right)\hat{k},-1200J \]

4. \[ 1600\left( N\times m \right)\hat{k},1200J \]

View Answer

Given:
- Dipole moment: \( \overrightarrow{p} = 3\hat{i} + 4\hat{j} \) C·m
- Electric field: \( \overrightarrow{E} = 0.4 \, \text{kN/C} \hat{i} = 400 \, \text{N/C} \hat{i} \)

Torque (\( \overrightarrow{\tau} \)):
\[
\overrightarrow{\tau} = \overrightarrow{p} \times \overrightarrow{E}
\]

\[
\overrightarrow{\tau} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
3 & 4 & 0 \\
400 & 0 & 0
\end{vmatrix} = \hat{k} \big(3(0) - 4(400)\big) = -1600\hat{k} \, \text{N·m}
\]

Potential Energy (\( U \)):
\[
U = -\overrightarrow{p} \cdot \overrightarrow{E}
\]
\[
U = -(3 \times 400 + 4 \times 0) = -1200 \, \text{J}
\]

Final Answer:
- Torque: \( -1600 \, \text{N·m} \hat{k} \)
- Potential energy: \( -1200 \, \text{J} \)

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Question 3: moderate

An electric dipole with dipole moment \( 2\times 10^{-9} \)Cm is aligned at 30º with the direction of a uniform electric field of magnitude \( 4\times 10^{4} NC^{-1}\). The magnitude of the torque acting on the dipole is :

1. \[ 2\times 10^{-5} Nm\]

2. \[ 2\times 10^{-4} Nm\]

3. \[ 4\times 10^{-4} Nm\]

4. \[ 4\times 10^{-5} Nm\]

View Answer

Given:
- Dipole moment: \( p = 2 \times 10^{-9} \, \text{C·m} \)
- Electric field: \( E = 4 \times 10^{4} \, \text{N/C} \)
- Angle: \( \theta = 30^\circ \)

Torque (\( \tau \)):
\[
\tau = pE \sin\theta
\]
\[
\tau = (2 \times 10^{-9})(4 \times 10^{4}) \sin 30^\circ
\]
\[
\tau = (8 \times 10^{-5}) \times \frac{1}{2} = 4 \times 10^{-5} \, \text{N·m}
\]

Final Answer:
\[
\tau = 4 \times 10^{-5} \, \text{N·m}
\]

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Question 4: moderate

A dipole of electric dipole moment p is placed in a uniform electric field of strength E. If θ is the angle between positive directions of p and E, then the potential energy of the electric dipole is largest when θ is :

1. π/4

2. π/2

3. π

4. Zero

View Answer

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

\[
U = -\mathbf{p} \cdot \mathbf{E} = -pE \cos\theta
\]

Here:
- \(p\) is the magnitude of the dipole moment,
- \(E\) is the magnitude of the electric field,
- \(\theta\) is the angle between \(\mathbf{p}\) and \(\mathbf{E}\).

The potential energy is largest when \(-\cos\theta\) is most positive, i.e., when \(\cos\theta = -1\). This happens at:

\[
\theta = \pi \ (\text{180°})
\]

At this angle, the dipole is aligned opposite to the electric field, and the potential energy is \(U = +pE\), its maximum value.

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Question 5: moderate

A point Q lies on the perpendicular bisector of an electric dipole of dipole moment p. If the distance of Q from the dipole is r (much larger than the size of the dipole) then electric field at Q is proportional to :

1. \(p^{-1} \)and \( r^{-2}\)

2. p and \( r^{-2}\)

3. \(p^{2}\) and \(r^{-3}\)

4. p and \( r^{-3}\)

View Answer

For a point \( Q \) on the perpendicular bisector of an electric dipole (distance \( r \) from the center, where \( r \gg \text{dipole length} \)):

1. Electric Field on Perpendicular Bisector: The electric field \( E \) at a point on the perpendicular bisector of a dipole is given by:
\[
E \propto \frac{p}{r^3}
\]

2. Dependence:
- Directly proportional to the dipole moment \( p \).
- Inversely proportional to \( r^3 \).

Answer:
The electric field at \( Q \) is proportional to:
\[
p \quad \text{and} \quad r^{-3}
\]

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