Induced EMF and angular velocity

[bon]

Homework Statement

A circular coil of area A with N turns rotates at an angular velocity w about an axis perpendicular to its own axis and to a uniform magnetic field B. Find an expression for the induced EMF in the coil in terms of B, N, A, w and time t. What is the orientation of the coil when the EMF is maximum?

Homework Equations

The Attempt at a Solution

So last part first - am i right in thinking that the EMF is max when the axis of the coil is parallel to the B-field?

Also I get EMF = NBAw/2pi..where does t come in?

I get this by Emf = dphi/dt

phi = NB dA/dt

dA/dt = A/(2pi/w)

Where does the t come in?

Thanks!

 

[Delphi51]

Take a look at a simulation:
http://www.sciencejoywagon.com/physicszone/otherpub/wfendt/generatorengl.htm

am i right in thinking that the EMF is max when the axis of the coil is parallel to the B-field?

No, the axis of the coil is always perpendicular to the magnetic field.

You can see that the EMF varies sinusoidally with time, so you need a cos(wt) or sin(wt) in your answer. The usual approach is to begin with the high school formula EMF = L*v*B, where v is the component of the velocity perpendicular to the magnetic field.

 

[bon]

Thanks but the app doesn't work on my computer :(

Please can you explain how to work this out? I don't see why the EMF varies with time?

 

[MisterX]

The induced EMF will approximately equal in magnitude to the rate of change of flux through one loop times the number of turns. You should know Faraday's Law of Induction.

You could express the flux through the cross sectional area A as it rotates.

This will be an integral of the flux density (B field) across the the surface with area A.

The flux is the integral of the B field over area, which is like summing at every point the dot product of the B field and the normal vector of the surface with area A. The B field is constant in one direction. So this integral is easy to calculate.

So, to solve the problem:
1. Express the normal vector as a function of time
2. Get the dot product of the normal vector with the constant B field
3. Multiply by the area
4. Take the derivative w.r.t. time and multiply by negative one to get the emf as a function of time to get the emf through one loop. Multiply by N to get the total emf.
5. Find what direction the normal vector from step 1 is facing when the emf is maximum.A quicker way is just figure out the phase relationship, knowing the flux will be a sine wave that has a maximum when the B field is parallel to the coil axis. The emf will be negative one times the derivative of that sine wave, so the emf will have a certain phase shift relative to the first sine wave.

 

[Delphi51]

That simulation only requires Java. A "get Java" button should appear in your browser. You pretty much need Java to view a lot of websites. In my opinion, the simulation is a vital part of your education in EM! Another thing I would certainly do is get a couple of feet of heavy wire (coat hanger perhaps), make a loop and play with it. Hold your hand in the appropriate position to use the hand rule to find the direction of the force on the electrons in each of the four sides of the loop. Later when you do Lenz' law, you will also use the hand rule to find the force opposing the current resulting from the first force.

There is a still picture here:
http://zebu.uoregon.edu/1999/ph161/l3.html
Scroll down a couple of screenfulls to see it.
You really need a diagram like this to get the sine or cosine into your formula. In MisterX's flux approach, you need one of these trig functions to get the reduced flux through the loop as the loop turns away from being perpendicular to the flux. Or in the L*v*B formula, the component of v perpendicular to B will have a sine or cosine of the angle. Of course the angle changes with time in both approaches.

 

[bon]

The induced EMF will approximately equal in magnitude to the rate of change of flux through one loop times the number of turns. You should know Faraday\'s Law of Induction.

You could express the flux through the cross sectional area A as it rotates.

This will be an integral of the flux density (B field) across the the surface with area A.

The flux is the integral of the B field over area, which is like summing at every point the dot product of the B field and the normal vector of the surface with area A. The B field is constant in one direction. So this integral is easy to calculate.

So, to solve the problem:
1. Express the normal vector as a function of time
2. Get the dot product of the normal vector with the constant B field
3. Multiply by the area
4. Take the derivative w.r.t. time and multiply by negative one to get the emf as a function of time to get the emf through one loop. Multiply by N to get the total emf.
5. Find what direction the normal vector from step 1 is facing when the emf is maximum.


A quicker way is just figure out the phase relationship, knowing the flux will be a sine wave that has a maximum when the B field is parallel to the coil axis. The emf will be negative one times the derivative of that sine wave, so the emf will have a certain phase shift relative to the first sine wave.

Thanks so much! This was extremely helpful. The answers I got are:

Total EMF = NABwsinwt

EMF maximum when the normal is perp to the B field.

Are these right?

Thanks again.

 

[Delphi51]

Looks good to me!