Electrostatics And Work Done By Electric Field

To solve this problem, we need to calculate the forces experienced by the charge particle due to each dipole and set the net force to zero for equilibrium.

Step 1: Understanding the Setup

We have two dipoles:

• Dipole 1: Charge configuration and , along the axis where the charge is placed at a distance .
• Dipole 2: Charge configuration and , along the equatorial plane where the charge is also at a distance .

Both dipoles have a separation distance .

Step 2: Force Calculation on Dipole Axial and Equatorial Points

The formula for the electric field due to a dipole at a point along its axial line is:

And along the equatorial line is:

where is the dipole moment.

Step 3: Net Force on Charge Q

The force on charge due to Dipole 1 (axially) is in the direction of the dipole:

The force on charge due to Dipole 2 (equatorially) is perpendicular to the axis and in the plane of the equator:

Step 4: Condition for Zero Net Force

For zero net force, the axial and equatorial forces must be equal:

Simplifying, we get:

(which means the terms must balance others, implying setups allow equatorial contribution equally with axial).

This satisfies the equilibrium if:

Conclusion

Thus, the correct answer ensuring no net force on charge is:

The question involves calculating the work done on an electric dipole when it is rotated from its equilibrium position through in a uniform electric field. The key concepts here are the electric dipole moment and the potential energy in an external electric field.

Conceptual Approach:
An electric dipole consists of two equal and opposite charges separated by a distance. When placed in a uniform electric field, the dipole experiences torque which tries to align it along the field direction. The work done on the dipole when rotated in the electric field is given by changing its potential energy.

Formulas Required:

• The dipole moment is given by: , where is the charge and is the distance vector between the charges.
• The potential energy of a dipole in an electric field is: , where is the angle between and .
• Work done when rotating the dipole from to is: .

Calculation:

1. Calculate the dipole moment :
   The charges are and , and the distance between them is from to , which is 18 cm or 0.18 m.
   
2. Calculate Initial Potential Energy :
   
3. Calculate Final Potential Energy :
   
4. Calculate the Work Done:
   

Therefore, the work done on the dipole when rotated by is 14.4 mJ.