An electric dipole in a uniform electric field has potential energy U. The magnitude of the torque acting on the dipole due to the field is `tau`.

Options

A. U is minimum and `tau` is zero when the dipole is perpendicular to the field.
B. U is zero and `tau` is a maximum when the dipole is perpendicular to the field.
C. U is a minimum and `tau` is a maximum when the dipole is perpendicular to the field.
D. U is a minimum and `tau` is zero when the dipole is anti-parallel to the field.

Answer

To solve the problem regarding the electric dipole in a uniform electric field, we need to derive the expressions for the torque (\( \tau \)) and potential energy (\( U \)) of the dipole. Here’s a step-by-step solution:

### Step 1: Understanding the Electric Dipole
An electric dipole consists of two equal and opposite charges (+q and -q) separated by a distance \( d \). The dipole moment \( \vec{p} \) is defined as:
\[ \vec{p} = q \cdot d \]
where \( d \) is the vector pointing from the negative charge to the positive charge.

### Step 2: Torque on the Dipole
When placed in a uniform electric field \( \vec{E} \), the dipole experiences a torque \( \tau \). The torque can be expressed as:
\[ \tau = \vec{p} \times \vec{E} \]
The magnitude of the torque is given by:
\[ \tau = pE \sin(\theta) \]
where \( \theta \) is the angle between the dipole moment \( \vec{p} \) and the electric field \( \vec{E} \).

### Step 3: Potential Energy of the Dipole
The potential energy \( U \) of the dipole in the electric field is given by:
\[ U = -\vec{p} \cdot \vec{E} = -pE \cos(\theta) \]
This indicates that the potential energy depends on the orientation of the dipole in the electric field.

### Step 4: Analyzing the Conditions
1. **When \( \theta = 0^\circ \)** (dipole aligned with the field):
- \( \tau = 0 \) (no torque)
- \( U = -pE \) (minimum potential energy)

2. **When \( \theta = 90^\circ \)** (dipole perpendicular to the field):
- \( \tau = pE \) (maximum torque)
- \( U = 0 \) (potential energy is zero)

3. **When \( \theta = 180^\circ \)** (dipole anti-aligned with the field):
- \( \tau = 0 \) (no torque)
- \( U = pE \) (maximum potential energy)

### Conclusion
- The potential energy \( U \) is minimum when the dipole is aligned with the electric field (\( \theta = 0^\circ \)).
- The torque \( \tau \) is maximum when the dipole is perpendicular to the electric field (\( \theta = 90^\circ \)).
- The potential energy \( U \) is maximum when the dipole is anti-aligned with the electric field (\( \theta = 180^\circ \)).

### Final Answer
Thus, the relationship between the potential energy \( U \) and the torque \( \tau \) in an electric dipole in a uniform electric field is summarized as follows:
- \( U \) is minimum and \( \tau \) is maximum when the dipole is perpendicular to the electric field.
- \( U \) is maximum and \( \tau \) is zero when the dipole is anti-parallel to the electric field.