The dimensional formula of self-inductance is . . . . . . .

Two solenoids have identical geometrical construction and the same number of turns,but one is made of thick wire and the other of thin wire. Which of the following quantities are different for the two solenoids?
$(a)$ Self-inductance.
$(b)$ Rate of Joule heating if the same current flows through them.
$(c)$ Magnetic potential energy if the same current flows through them.
$(d)$ Time constant.

An air-cored solenoid with length $30 \; cm$,area of cross-section $25 \; cm^{2}$ and number of turns $500$,carries a current of $2.5 \; A$. The current is suddenly switched off in a brief time of $10^{-3} \; s$. How much is the average back $emf$ induced (in $V$) across the ends of the open switch in the circuit? Ignore the variation in magnetic field near the ends of the solenoid.

$A$ hollow cylinder has length $l$,radius $r$,and thickness $d$,where $l >> r >> d$,and is made of a material with resistivity $\rho$. $A$ time-varying current $I$ flows through the cylinder in the tangential direction. Assume the current is always uniformly distributed along the length of the cylinder. The cylinder is fixed so that it cannot move; assume that there are no externally generated magnetic fields during the time considered for the problems below. Assume current at $t = 0$ to be $I_0$. What is the current $I(t)$ for $t > 0$?

$A$ solenoid of length $l$ metre has self-inductance $L$ henry. If the number of turns is doubled,its self-inductance becomes:

$A$ coil of wire of radius $r$ has $600$ turns and self-inductance of $108 \ mH$. The self-inductance of a coil with the same radius and $500$ turns is (in $mH$)