Question

Two solenoids have identical geometrical construction 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 goes through them
(c) magnetic field energy if the same current goes through them
(d) time constant if one solenoid is connected to one battery and the other is connected to another battery.

Answer 1

(b) rate of Joule heating if the same current goes through them
(d) time constant if one solenoid is connected to one battery and the other is connected to another battery

Because the solenoids are identical, their self-inductance will be the same.
Resistance of a wire is given by
$R=\rho \frac{l}{A}$
Here,
l = Length of the wire
A = Area of cross section of the wire
ρ = Resistivity of the wire

Because ρ and l are the same for both wires, the thick wire will have greater area of cross section and hence less resistance than the thin wire.
$\Rightarrow R_{thick}<R_{thin}$

The time constant for a solenoid is given by
$\tau =\frac{L}{R}$
$\therefore {\tau }_{thick}>{\tau }_{thin}$
Thus, time constants of the solenoids would be different if one solenoid is connected to one battery and the other is connected to another battery.

Also, because the self-inductance of the solenoids is the same and the same current flows through them, the magnetic field energy given by $\frac{1}{2}L{i}^2$ will be the same.
Power dissipated as heat is given by
$P=i^{2}R$
i is the same for both solenoids.
$\therefore P_{thick}<P_{thin}$
Because the resistance of the coils are different, the rate of Joule heating will be different for the coils if the same current goes through them.