A dipole of dipole moment `vec(P)` is placed in uniform electric field `vec(E)`, then torque acting on it is given by

Options

A. `vec(tau)=vec(P).vec(E)`
B. `vec(tau)=vec(P)xx vec(E)`
C. `vec(tau)=vec(P)+vec(E)`
D. `vec(tau)=vec(P)-vec(E)`

Answer

To find the torque acting on a dipole moment \(\vec{P}\) placed in a uniform electric field \(\vec{E}\), we can follow these steps:

### Step-by-Step Solution:

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

2. **Forces on the Dipole**:
When the dipole is placed in an electric field \(\vec{E}\), the positive charge experiences a force \(\vec{F}_+ = q\vec{E}\) in the direction of the field, while the negative charge experiences a force \(\vec{F}_- = -q\vec{E}\) in the opposite direction.

3. **Torque Calculation**:
The net force on the dipole is zero, but these forces create a torque \(\vec{\tau}\) about the center of the dipole. The torque can be calculated using the formula:
\[ \vec{\tau} = \vec{r} \times \vec{F} \]
where \(\vec{r}\) is the position vector from the center of the dipole to the point of application of the force.

4. **Finding the Perpendicular Distance**:
The perpendicular distance \(R\) from the line of action of the force to the pivot point (the center of the dipole) can be expressed in terms of the angle \(\theta\) that the dipole makes with the electric field:
\[ R = d \sin \theta \]
where \(d\) is the length of the dipole.

5. **Expression for Torque**:
The torque due to the force on the positive charge is:
\[ \tau = F_+ \cdot R = (qE)(d \sin \theta) \]
The torque can also be expressed in terms of the dipole moment:
\[ \tau = (q \cdot d) E \sin \theta = \vec{P} \cdot \vec{E} \sin \theta \]

6. **Final Vector Form**:
The torque acting on the dipole in vector form is given by:
\[ \vec{\tau} = \vec{P} \times \vec{E} \]
This indicates that the torque is a vector quantity and is perpendicular to the plane formed by \(\vec{P}\) and \(\vec{E}\).

### Final Result:
Thus, the torque acting on the dipole in a uniform electric field is given by:
\[ \vec{\tau} = \vec{P} \times \vec{E} \]