A conducting loop is formed with two springs, each with a spring constant, k=2N/

Question

A conducting loop is formed with two springs, each with a spring constant, k=2N/m, and a rod of length, I=30cm, and mass, m=0.020kg. A uniform magnetic field of 0.4T is directed perpendicular to the plane of the loop. At time, t=0s, the rod is released with the springs extended by deltaX=10cm.

a) write an expression for the induced emf in terms of the magnetic field, the legth of the rod, the maximum velocity of the rod, the displacement of the rod, and the maximum displacement of the rod. (Hint: treat the system as a harmonic oscillator)

b) what is the maximum value of the emf, and after how much time after the rod is released does the maximum emf first occur?

Explanation / Answer

This is a pretty complicated problem... and if your prof wrote this problem, it might be more complicated than he or she realizes.

What he or she is giving you is a damped harmonic oscillator, which is described by the equation:

0 = a + (b/2m) v + w^2 x   

In your case w^2 = 2k/m and b = L^2 B^2 / R (R is the resistance of the loop, which is not given.)

I think your prof neglected to give you the resistance of the loop because he or she forgot that there would be a damping force (i.e. a resistive force acting on the loop that is proportional to the speed of the bar) due to the force of the magnetic field acting on the induced current. If the problem had been given as a non-conducting loop, you would not have this issue.

Anyway... you might want to bring this to the attention of your prof. Basically what you want for part a is:

induced emf = B L v where v is the speed of the rod

Position of the rod is given by x = A [e^(-b/2m)t ] cos(Ct)   

where C = sqrt( 2k/m - (b/2m)^2 )

Take the first derivative of that expression to get v. It's pretty ugly.

Your best course of action is to ask the prof (send them an email if you like):

"Did you mean to make this a conducting loop? Because if it's conducting, wouldn't there be a damping force acting on the rod due to the magnetic force acting on the induced current? So wouldn't the oscillator act as a damped oscillator, which would really complicate the expression for the speed of the rod? And wouldn't we need to include the resistance of the rod in that expression, since it affects the current, which affects the damping force? Or did you mean to make it a non-conducting loop?"

See how they respond. I'm sure you'll find I'm right, that they didn't mean to make it this complicated. Profs make mistakes sometimes. I know from personal experience.

If the loop is non-conducting, this complication disappears.

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