How to calculate the mutual inductance?

[ayoubster]

Homework Statement

A long solenoid has a radius of 3 cm, 3000 turn per meter, and carries a current I = IOcos(ωt), where Io is 0.25 A and ω is 628 s−1 . It is placed through a circular loop of wire, radius 5 cm, which has resistance 100 Ω. The magnetic field in a solenoid is B = µonI.
(a) Find the mutual inductance of the solenoid and wire loop.
(b) Find the emf induced in the loop as a function of time, and the peak current which will flow as a result.
(c) Find the maximum electric field induced by the solenoid at the wire loop’s distance from the axis.

Homework Equations

M = (Nφ) / I
φ = ∫BA
ε = M(dI/dt)
ε = ∫Eds = dφ/dt

The Attempt at a Solution

I can integrate to find the flux of the solenoid, but I don't have the current of the loop. I can do the opposite and find the flux of the loop since I have the current of the solenoid, but I don't have the magnetic field, I'm stuck on (a)

b) Taking the derivative of I gives Iωsin(ωt), however I don't have the mutual inductance to calculate it

c) Kind of lost on this one

 

[TSny]

WELCOME TO PF!

The Attempt at a Solution

I can integrate to find the flux of the solenoid, but I don't have the current of the loop. I can do the opposite and find the flux of the loop since I have the current of the solenoid, but I don't have the magnetic field

I'm not understanding your difficulty. Make sure you are clear on the exact interpretation of the symbols in M = (Nφ) / I.

 

[ayoubster]

Sorry, the text field isn't easy to work with as its my first time. The symbols in that equations correspond to

N = to number of turns in the coil per meter
Φ = to the magnetic flux of the solenoid
I = to the current in the loop

Mutual inductance can be calculated by using the current in the solenoid and the flux of the loop, however that would be impractical as I don't have the magnetic field strength of the loop. How do I go about finding the mutual inductance in terms of the current in the loop? The equation corresponds to that but the second question asks what the peak current in the loop is, so I am fairly confused about this.

 

[TSny]

The symbols in that equations correspond to

N = to number of turns in the coil per meter
Φ = to the magnetic flux of the solenoid
I = to the current in the loop

This isn't quite right.

Often, the formula for $M$ is written with subscripts as $M_{21} = \frac{N_2 \Phi_{21}}{I_1}$. The subscripts help guide the correct interpretation of the symbols.

In general, you have two "coils" labeled 1 and 2. $N_2$ is the total number of turns in coil 2, not the number of turns per meter. $\Phi_{21}$ is the magnetic flux through one turn of coil 2 due to the field produced by the current in coil 1.

Suppose you let coil 1 be the solenoid and coil 2 be the loop. How would you set up $M_{21} = \frac{N_2 \Phi_{21}}{I_1}$?