Circular loop in magnetic field

[johnknee]

Homework Statement

A circular single loop of wire with a diameter of 20.0 cm lies in the plane of the paper in a region of space that contains a 1.75 T magnetic field pointing out of the paper.

a) Determine the magnetic flux through this loop.

b) If the diameter of the loop changes from 20.0 cm to 10.0 cm in 0.25 seconds, what is the direction of the induced current, clockwise or counter-clockwise as seen from above? Explain your answer.

c) What is the magnitude of the induced emf that results from the diameter change in part b)?

d) What is the magnitude of the induced emf if this new smaller loop (d = 10.0 cm) is now rotated about an axis along the diameter by 90 deg in 1.50 seconds in the given magnetic field so that its normal now lies in the plane of the paper?

Homework Equations

Flux = B*A*Cos(theta)
Induced emf = (change in flux)/change in time
A = pi*r^2

The Attempt at a Solution

part a) I have that A = pi*0.1^2 = 0.0314m^2
Flux = B*A*cos(0) = (1.75T)(0.0314m^2)cos(0) = 0.0550 Wb

part b) I said that it is clockwise because since B is increasing and out of the page, lenz's law suggests that it should therefore have B go into the page to oppose the change. By using right hand rule, fingers curl clockwise when thumb points into the page.

part c) I used induced emf = delta flux/time
induced emf = (1.75*(pi*0.05^2 - pi*0.1^2))/0.25s = 0.165 V

part d) I am not sure what angle to use for this. I think that I would have to find flux using the formula B*A*cos(theta). Any hint would be much appreciated!

*Also, am I doing the other 3 parts correctly?

*edit for part d
I tried doing it with 0 deg again and i did B*A*cos0 = 1.75*(pi*0.05^2)*cos0 = 0.0137 Wb
then i did
induced emf = (0.0137wb)/(1.5s) = 0.00913 V. Is this how I am supposed to do part d?

 

[johnknee]

Edit for d) Would induced emf be 0 because since the loop is rotated 90 degrees, the normal now lies perpendicular to the direction of magnetic field. Flux would be 0 when angle is 90 degrees so emf = 0?

 

[haruspex]

Edit for d) Would induced emf be 0 because since the loop is rotated 90 degrees, the normal now lies perpendicular to the direction of magnetic field. Flux would be 0 when angle is 90 degrees so emf = 0?

The emf is created by the motion, not any particular position. Your solutions look fine to me, though I'm not very happy with the questions. They should be asking for the average induced emfs. Depending on exactly how the motions are performed, the emfs are likely to vary over time. In c, if the radius is reduced at a constant rate then the area will reduce faster at first then more slowly. Similarly for d if the loop is rotated at constant angular speed.

 

[johnknee]

part e) What would be the maximum emf created if one were to create a simple generator with this smaller loop and spin it about an axis along the diameter at a rate of 65.0 rev/s in the given magnetic field?

For this part, am I using the right approach?
Max induced voltage = N*B*A*Angular Velocity.
Angular velocity = 65 rev/s * 2pi rad/1 rev = 408.4 rad/s.

Max induced voltage = (1)*(1.75 T)*(pi*0.05^2 m^2)*(408.4 rad/s) = 5.61 V

 

[haruspex]

part e) What would be the maximum emf created if one were to create a simple generator with this smaller loop and spin it about an axis along the diameter at a rate of 65.0 rev/s in the given magnetic field?

For this part, am I using the right approach?
Max induced voltage = N*B*A*Angular Velocity.
Angular velocity = 65 rev/s * 2pi rad/1 rev = 408.4 rad/s.

Max induced voltage = (1)*(1.75 T)*(pi*0.05^2 m^2)*(408.4 rad/s) = 5.61 V

Where are you getting that formula from? A formula always has context, i.e. the physical set up in which the formula applies, and exactly what the variables represent. In what context is that formula appropriate?

 

[johnknee]

For part e) I got that formula from the textbook from the generator section. It said that max emf = NBA(angular velocity). I am not quite sure if I would apply that equation here in this specific question.

 

[haruspex]

For part e) I got that formula from the textbook from the generator section. It said that max emf = NBA(angular velocity). I am not quite sure if I would apply that equation here in this specific question.

Ok, but the textbook must quote that equation in respect of a specific arrangement. Does that arrangement match the problem in this thread, i.e., a loop of area A being rotated about an axis that is coplanar with the loop and perpendicular to the field, at constant angular velocity?
I'm not saying the equation is wrong here, but you should apply equations on the basis of logic, not mere hope.