A point charge of mass is suspended vertically by a string of length. A point dipole of dipole moment vector is now brought towards from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is where is the acceleration due to gravity, then the value of is _________ .(Note that for three coplanar forces keeping a point mass in equilibrium, F/sin θ is the same for all forces, where F is any one of the forces and θ is the angle between the other two forces)
Step 1. Given data:
The charge is and mass is
Length of string
Dipole moment
Work done
Step 2. Dipole moment in the field of charge
From the diagram,
where is the angle when charge is moved due to dipole moment
is the other two angles of the triangle thus formed
Work done in moving charge is,
where is the vertical displacement in charge, is the charge, is the displacement in charge position
and , is the permittivity
Step 3. Equating the coplanar forces using sign law, we get
where is the tension in the string, is mass of charge, is acceleration due to gravity, is the charge, and is the displacement in charge position
From diagram in triangle BDC,
Putting in equation
So, the work done is,
Hence, the value of is .