Two solenoids hav 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.

The two solenoids are identical. Therefore, the self-inductance of the two solenoids is the same.

The formula used: energy stored in the inductor is given by

Where U is the Magnetic energy

L is the self-Inductance

I is the Current

The current in both solenoids is the same. That implies magnetic field energy is also the same.

The power dissipated as heat is given by

Where P is the power

I is the current

R is the Resistance

The time constant is given by

Where t is the time constant

L is the self-Inductance

R is resistance is given by

Where ρ is the resistivity

l is the length of the wire

A is the Area of cross section

The two solenoids are differed by the Area of the cross section A.

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Thus, the time constant and Joule heat is different for the two solenoids. And self-inductance and magnetic field are the same for the both.