In a uniform electric field E=(x i^+y j^) N/C an electric dipole p= (a i^+ bj^) C-m is rotated clockwise by 90° in its plane .find work performed

To determine the work performed when an electric dipole is rotated in a uniform electric field, we need to consider the relationship between the dipole moment, the electric field, and the angle through which the dipole is rotated. Let's break this down step by step.

Understanding the Electric Dipole and Electric Field

An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment, represented as **p**, is a vector quantity defined as the product of the charge and the distance between the charges. In your case, the dipole moment is given as **p = (a i + b j) C·m**.

The electric field **E** is uniform and given as **E = (x i + y j) N/C**. This means that the electric field varies with position in the x and y directions.

Work Done on the Dipole

The work done **W** on an electric dipole when it is rotated in an electric field can be calculated using the formula:

W = -p · E (cos θ_f - cos θ_i)

Where:

• **p** is the dipole moment vector.
• **E** is the electric field vector.
• **θ_f** is the final angle of the dipole with respect to the field.
• **θ_i** is the initial angle of the dipole with respect to the field.

Calculating the Initial and Final Angles

Assuming the dipole is initially aligned with the electric field, the initial angle **θ_i** is 0°. After rotating the dipole clockwise by 90°, the final angle **θ_f** becomes 90°.

Substituting Values into the Work Formula

Now, we can substitute the angles into the work formula:

W = -p · E (cos 90° - cos 0°)

Since **cos 90° = 0** and **cos 0° = 1**, the equation simplifies to:

W = -p · E (0 - 1) = p · E

Calculating the Dot Product

Next, we need to compute the dot product **p · E**. Given:

**p = (a i + b j)**

**E = (x i + y j)**

The dot product is calculated as:

p · E = (a i + b j) · (x i + y j) = ax + by

Final Expression for Work Done

Substituting this back into our work formula gives:

W = ax + by

This expression represents the work done on the dipole when it is rotated 90° in the uniform electric field. The work depends on the components of the dipole moment and the electric field at the position where the dipole is located.

Summary

In summary, the work performed when rotating an electric dipole in a uniform electric field is given by the expression **W = ax + by**, where **a** and **b** are the components of the dipole moment, and **x** and **y** are the components of the electric field. This relationship highlights how the orientation of the dipole in the electric field influences the work done during the rotation.