Electric Fields And Gauss Law

Step 1: Understanding the relationship.
The electric field intensity is the rate of change of electric potential with respect to distance .
Step 2: Formula for electric field intensity.
The formula for the electric field intensity in terms of potential gradient is: where is the change in potential and is the change in distance. The negative sign indicates that the electric field points in the direction of decreasing potential.
Step 3: Conclusion.
The electric field intensity is the negative of the rate of change of electric potential with respect to distance.

Step 1: Formula for Torque.
The torque experienced by an electric dipole in a uniform electric field is given by: where is the dipole moment, is the electric field, and is the angle between the dipole moment and the electric field.
Step 2: Calculate Dipole Moment.
The dipole moment is given by: where is the charge and is the separation between the charges. Substituting the values:
Step 3: Maximum Torque.
The maximum torque occurs when , i.e., when . The maximum torque is then: Substituting and :
Step 4: Work Done.
The work done to rotate the dipole from to is given by the change in potential energy: Substituting and : Substituting and : Thus, the work done is .