Magnetic Fields Shrinking Radius

[SWFanatic]

Homework Statement

An elastic conducting material is stretched into a circular loop of 20.0 cm radius. It is placed with its plane perpendicular to a uniform 0.900 T magnetic field. When released, the radius of the loop starts to shrink at an instantaneous rate of 78.0 cm/s. What emf is induced in the loop at that instant?

Homework Equations

The Attempt at a Solution

I am very lost in this section. Any help would be appreciated...just a starting point please! Thanks

 

[Mindscrape]

What is the definition of flux? How does EMF relate to flux?

 

[Moetasim]

I would approach this problem by first understanding the concept of electromagnetic induction. When a conductor, such as the elastic material in this case, moves through a magnetic field, it experiences a change in magnetic flux and an electromotive force (emf) is induced in the conductor. This emf is given by the equation emf = -N(dΦ/dt), where N is the number of turns in the conductor and dΦ/dt is the rate of change of magnetic flux.

In this problem, we are given the radius of the circular loop, the rate at which the radius is shrinking, and the strength of the magnetic field. We can use this information to calculate the rate of change of magnetic flux, dΦ/dt. We know that the magnetic flux through a loop is given by Φ = BA, where B is the magnetic field strength and A is the area of the loop. Since the area of the loop is changing as the radius shrinks, we can use the equation A = πr^2 to find the area at any given time. Therefore, the rate of change of magnetic flux can be calculated as dΦ/dt = B(dA/dt) = 2πBr(dR/dt), where R is the radius of the loop at any given time.

Now, we have all the necessary information to calculate the emf induced in the loop at the given instant. We know the number of turns in the loop (which is not explicitly given in the problem, but we can assume it to be one), the magnetic field strength, and the rate of change of magnetic flux. Plugging these values into the equation for emf, we get emf = -N(dΦ/dt) = -2πBr(dR/dt). Substituting the given values, we get emf = -(1)(0.900 T)(2π)(20.0 cm)(78.0 cm/s) = -354.4 mV.

Therefore, at the instant when the radius of the loop is shrinking at a rate of 78.0 cm/s, an emf of 354.4 mV is induced in the loop. This emf will cause a current to flow in the loop, which can be used to do work or power a device connected to the loop.