Electric And Magnetic Fields

Step 1: Understanding equipotential surfaces.
An equipotential surface is a surface on which every point has the same electric potential. The electric potential at each point on an equipotential surface is constant.
Step 2: The impossibility of intersection.
If two equipotential surfaces were to intersect, there would be a contradiction. At the point of intersection, the potential would have to be two different values at the same point, which is not possible. Therefore, two equipotential surfaces cannot intersect.
Step 3: Conclusion.
Thus, two equipotential surfaces cannot intersect because it would imply that a point has two different potentials, which is physically impossible.

Step 1: Understanding the effect of a dielectric.
The capacitance of a capacitor is given by the equation: where is the permittivity of free space, is the area of the plates, and is the separation between the plates.
Step 2: Effect of introducing a dielectric.
When a dielectric material with dielectric constant is inserted between the plates of a capacitor, the capacitance increases by a factor of . The new capacitance becomes: Thus, the capacitance increases when a dielectric is inserted.
Step 3: Conclusion.
Therefore, when a dielectric is inserted between the plates of a capacitor, its capacitance increases, corresponding to option (A).

Step 1: Electric field outside the hollow sphere.
We apply Gauss' law to calculate the electric field at a point outside the hollow sphere. According to Gauss' law: where is the electric flux through a closed surface, is the charge enclosed by the surface, and is the permittivity of free space.
For a point outside the hollow sphere, we choose a spherical Gaussian surface with radius (where is greater than the radius of the shell). The charge enclosed by this surface is the total charge on the shell. Because of the symmetry of the situation, the electric field is radially symmetric and has the same magnitude at every point on the Gaussian surface. Therefore, the flux is: Solving for , we get: Thus, the electric field outside the hollow sphere behaves as if the entire charge were concentrated at the center of the sphere, similar to the electric field due to a point charge.
Step 2: Electric field inside the hollow sphere.
Now, consider the electric field at a point inside the hollow sphere. Since there is no charge inside the spherical shell (the charge is distributed on the outer surface), the charge enclosed by a Gaussian surface inside the shell is zero. According to Gauss' law: Thus, the electric field inside the hollow sphere is zero. This result is a consequence of the symmetry of the situation, where the charges on the shell produce no electric field at points inside the shell.
Step 3: Conclusion.
Thus, the electric field intensity outside the shell is: and the electric field inside the shell is:

Step 1: Capacitor and Potential Energy.
The potential energy stored in a capacitor is given by the formula: where is the potential energy, is the capacitance of the capacitor, and is the potential difference across the plates of the capacitor.
Step 2: Derivation.
The energy stored in the capacitor can also be calculated as the work done to move a small charge from one plate to the other against the potential difference . The total work done in charging the capacitor from 0 to is the potential energy stored in the capacitor: where is the total charge and . Substituting this into the integral: Since , we get: Step 3: Conclusion.
Thus, the expression for the potential energy stored in a charged capacitor is: