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.

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: \( -1200 \, \text{J} \)

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

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.

For an electric dipole in a uniform electric field, the torque \( \tau \) on the dipole is given by:

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

where:
- \( p \) is the dipole moment,
- \( E \) is the electric field strength,
- \( \theta \) is the angle between the dipole and the electric field.

Analysis:

1. Torque Variation:
- The torque \( \tau \) is maximum when \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \dots \) (odd multiples of \( \frac{\pi}{2} \)), as \( \sin \theta = \pm 1 \).
- The torque \( \tau \) is zero when \( \theta = 0, \pi, 2\pi, \dots \), as \( \sin \theta = 0 \).

2. Graph Interpretation:
- The correct graph of \( \tau \) versus \( \theta \) will have positive and negative peaks at \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \dots \), and cross zero at \( \theta = 0, \pi, 2\pi, \dots \).

Conclusion:

The correct answer is (b) as it represents the sinusoidal variation of torque with respect to \( \theta \) with alternating positive and negative values, matching the behavior of \( \tau = pE \sin \theta \).

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:
\[
p \quad \text{and} \quad r^{-3}
\]

The electric potential \( V \) at a point \( P \) due to an electric dipole with dipole moment \( \overrightarrow{p} \) is given by:

\[
V = \frac{k \, \overrightarrow{p} \cdot \overrightarrow{r}}{r^3}
\]

Explanation:
1. Dipole Moment: The dipole moment \( \overrightarrow{p} = q \cdot d \), where \( q \) is the charge and \( d \) is the separation between charges.

2. Position Vector \( \overrightarrow{r} \): This is the vector from the center of the dipole to the point \( P \).

3. Dot Product: The potential depends on the angle between \( \overrightarrow{p} \) and \( \overrightarrow{r} \), hence \( \overrightarrow{p} \cdot \overrightarrow{r} = p r \cos \theta \).

4. Result: The formula is:
\[
V = \frac{k (\overrightarrow{p} \cdot \overrightarrow{r})}{r^3}
\]

This matches the correct answer:
\[
V = \frac{k (\overrightarrow{p} \cdot \overrightarrow{r})}{r^3}
\]