A very long solenoid with a circular cross section and radius r1= 2.80 cm with n
ID: 1553081 • Letter: A
Question
A very long solenoid with a circular cross section and radius r1= 2.80 cm with ns= 220 turns/cm lies inside a short coil of radius r2= 4.70 cm and Nc= 24 turns.
If the current in the solenoid is ramped at a constant rate from zero to Is= 2.80 A over a time interval of 45.0 ms, what is the magnitude of the emf in the outer coil while the current in the solenoid is changing? Use Faraday's Law.
What is the mutual inductance between the solenoid and the short coil?
Now reverse the situation. If the current in the short coil is ramped up steadily from zero to Ic= 2.40 A over a time interval of 22.0 ms, what is the magnitude of the emf in the solenoid while the current in the coil is changing?
Explanation / Answer
r1 = 2.8*10^-2 m
ns = 220 turns/cm = 220*10^2 turns/m
r2 = 2.7*10^-2 m
Nc = 24
Is = 2.8 A
t = 45*0^-3 s
The magnitude of the EMF is given by E2 = M (Is)/t
The mutual inductance of a coil is given by M = 2/I1
Since the solenoid is enclosed by the outer loop, the flux though the outer loop will be the same as the flux through the solenoid.
Flux is given by = BA
The total flux in the coil is the number of turns in the coil multiplied by the flux in each turn.
Thus, M = (Nc2)/ I1 = (NcB1A1) /I1
M= (Ncµ0nsA1) /I1 = Ncµ0nsA1
M = Ncµ0ns(r1^2)
M = 24*8.85*10^-12*220*10^2 (3.14*2.8^2*10^-4)
M = 1.15*10^-10
The magnitude of the EMF is given by
E2 = M (Is)/t
E2 = - 1.15*10^-10 * 2.8/(45*10^-3)
E2 = - 71.6 * 10^-10
For part C
Ic = 2.4 A
t = 22 ms
Realizing that the mutual inductance M is a constant for a given configuration of wire, the EMF is simply given by the equation E2 = M Ic t