$(1)$ Suppose, a particle $($Volume: V$)$ with susceptibility X is placed at the center of a solenoid coil $($radius: R$,$ length L$,$ N turns$).$ Radius of the particle is much smaller than the radius of the coil. Analytically derive the change in voltage developed across the solenoid due to the presence of this particle when an ac current I $($peak$)$ flows through the solenoid at frequency f$.$ If you cannot get an analytical solution, provide steps as to how to solve this problem.

$(2)$ Now suppose the same particle is placed along the axis of the same solenoid coil at a distance D away from edge of the solenoid. Analytically derive the change in voltage developed across the solenoid due to the presence of this particle when an ac current I $($peak$)$ flows through the solenoid at frequency f$.$ If you cannot get an analytical solution, provide steps as to how to solve this problem.

Please look carefully at the questions and do not provide AI answers! I have got too many unreliable answers.