Finding RMS current in an AC solenoid

[chinye11]

Homework Statement

A 2000 turn solenoid of length 1.50 m and diameter 5.00 cm has a dc resistance of 3.0. It is
connected to a 50 Hz, 40.0 Volt (rms) ac supply. Find the rms current in the solenoid

Homework Equations

Vind = L di/dt
Vind = Induced EMF
Ohm's Law
V-Vind = IR
L = (4pi x 10^-7)n^2 Al

Where L=self inductance and l = length

The Attempt at a Solution

This is my first time using this website and i am not sure how to get in symbols, I will do my best to be clear

I started finding the inductance.

I then used Ohm's Law: V-Vind=IR

After mathematical manipulation I got I = V/R[1-e^(Rt/L)]

My question is as regards the frequency. It seems to have no effect on the system if you use my method yet if I try to find the induced E.M.F. using Faraday's Law on the magnetic flux you will find:

Vind= 0.05 Sin(theta) I(t)
Vind= 0.05 Sin (2 pi (f) t) I(t) from theta=wt

So I was wondering if someone could explain both the concept and the calculations of the effect of the frequency. Thank You.

 

[chinye11]

Just to clarify I am aware that the current is periodic and that the frequency can help in calculating max current and other such things however, I was wondering if it was of any use in this specific question.

 

[gneill]

You've effectively got a resistor and inductor in series, and it's driven by a 50Hz AC supply. If you were to calculate the effective impedance (Z) of the resistor+inductor at the given frequency, you could apply Ohm's law to find the current: I = V/Z. The magnitude of I is what you're after.

For an inductor the impedance varies in direct proportion to the frequency.

This presumes that you've been introduced to the concept of complex impedance...

 

[chinye11]

Thanks very much I hadnt covered impedance well. I have since gone through and understand it.

 

[rwisz]

The frequency of the AC supply does have an effect on the solenoid and the induced EMF. In your attempt at a solution, you correctly used Faraday's Law to calculate the induced EMF, which is given by Vind = N*dPhi/dt, where N is the number of turns in the solenoid and dPhi/dt is the rate of change of magnetic flux through the solenoid. This induced EMF is what causes the current to flow through the solenoid.

The frequency of the AC supply affects the rate at which the magnetic flux through the solenoid is changing. Since the AC supply has a frequency of 50 Hz, the magnetic flux through the solenoid is changing 50 times per second. This means that the induced EMF, and therefore the current, will also be changing 50 times per second.

In order to calculate the RMS current, you will need to use the formula I = V/R[1-e^(Rt/L)] for each time interval during one cycle of the AC supply. You can use the frequency to determine the time interval for one cycle, which is 1/50 seconds. Then, you can use this formula to calculate the current for each time interval and take the average of all the currents to get the RMS current.

In summary, the frequency of the AC supply affects the rate at which the induced EMF and current are changing, which is important in calculating the RMS current. I hope this helps clarify the concept and calculations for you.