An electric dipole of moment `vecp` is placed in a uniform electric field `vecE` . Then
(i) The torque on the dipole is `vecp xxvecE`
(ii) The potential energy of the system is `vecp.vecE`
(iii) The resultant force on the dipole is zero.

Options

A. (i), (ii) and (iii) are correct
B. (i) and (iii) are correct and (ii) is wrong
C. Only (i) is correct
D. (i) and (ii) are correct (iii) is wrong

Explanation

To solve the problem regarding the electric dipole placed in a uniform electric field, we will analyze the three statements provided in the question step by step.

### Step 1: Understanding the Electric Dipole
An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance \(2a\). The dipole moment \(\vec{p}\) is defined as:
\[ \vec{p} = q \cdot \vec{d} \]
where \(\vec{d}\) is the vector pointing from the negative charge to the positive charge.

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

### Step 3: Potential Energy of the Dipole
The potential energy \(U\) of the dipole in the electric field is given by the negative dot product of the dipole moment and the electric field:
\[ U = -\vec{p} \cdot \vec{E} \]
This indicates that the potential energy is minimized (most negative) when the dipole is aligned with the electric field.

### Step 4: Resultant Force on the Dipole
In a uniform electric field, the forces acting on the positive and negative charges of the dipole are equal in magnitude and opposite in direction. Therefore, the net force \(\vec{F}\) on the dipole is:
\[ \vec{F} = \vec{F}_+ + \vec{F}_- = q\vec{E} - q\vec{E} = 0 \]
Thus, the resultant force on the dipole is zero.

### Conclusion
Now, we can evaluate the three statements:
1. The torque on the dipole is \(\vec{p} \times \vec{E}\) - **True**.
2. The potential energy of the system is \(\vec{p} \cdot \vec{E}\) - **False** (it should be \(-\vec{p} \cdot \vec{E}\)).
3. The resultant force on the dipole is zero - **True**.

### Final Answer
The correct statements are:
- Statement 1 is correct.
- Statement 2 is incorrect.
- Statement 3 is correct.

Thus, the answer is that the first and third statements are correct, while the second is wrong.

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