A smaller solenoid is placed inside a larger solenoid so they have the same axis

Question

A smaller solenoid is placed inside a larger solenoid so they have the same axis (like concentric cylinders). The inner solenoid has radius r=0.014meters, N=1000turns, and resistance R=13Ohms. Its two leads are connected to an ammeter to form a circuit. The outer solenoid has n=23000turns per meter and carried a current that is increasing ata constant rate of dI/dt=0.5Amperes per second.
What is the current throught the ammeter? What is its direction?

Iinner =___A

Compared to the direction of current through the outer solenoid, the current through the inner solenoid moves in (choose one):

___the same direction.
___the opposite direction.

Explanation / Answer

The magnetic field inside the inner solenoid changes with time because of the change in the current through the outer solenoid. Now let I is the current in the outer solenoid, the magnetic field inside the solenoid is:

(B = mu_0I*n.), where n is number of turns per unit length in the outer solenoid, thus

(B = 23000 mu_0I.)

Thus, the flux through one turn of the inner solenoid is:

(phi_1 = Bpi r^2 = 23000mu_0I imes 3.1416(0.014)^2=14.16mu_0I.)

thus, rate of change of flux through 1000 turns is:

(rac{dphi_{total}}{dt}=1000 imes 14.16 imes 4pi imes 10^{-7}rac{dI}{dt}.)

Also, (rac{dphi}{dt}=-epsilon.)

and given (rac{dI}{dt}=0.5 A/s.)

where (epsilon.) is the total induced emf

thus emf induced is

(epsilon=-0.0089.)

also, current i in the solenoid is then,

(i=rac{epsilon}{R}=-6.8 imes 10^{-5} A.)

now, the direction of current will be opposite to the outer solenoid.

This is so because since the current increases in the outer solenoid, the flux through inner solenoid increases. Thus, to oppose this change in flux, the current in the inner solenoid should be in such a direction which decreases the magnetic flux

Thus, the current should be iin opposite direction that to the current in the outer solenoid.

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