# Homework Help: Induced EMF in Rectangular Loop of Changing Area in Uniform/Constant Magnetic Field

1. Aug 14, 2012

### nickmanc86

1. A rectangular wire loop (length 60 cm, width 40 cm) lies completely within a
perpendicular and uniform magnetic field of magnitude of 0.5 T. If the length of the loop
starts increasing at a rate of 20 mm/s at time t = 0, while the width is decreasing at the
same rate, what is the magnitude of the induced emf at time t = 4.0 s?

Multiple Choice
a) 6.8
b) 5.2
c) 3.6
d) 8.4
e) 10mv

2. $$\varepsilon = -N \Delta \Phi /\Delta t$$ and $$\Phi = B*A$$

3. I took the simple approach using the given time I took 20mm/s * 4.0 s =.08m which I then add to the length and subtract from the width. A(f)=.218, A(i)=.24, t(f)=4 (i)=0 take the $\Delta A/\Delta t$ and multiply by B=.5 and get an answer of 2.8mV

I have some other ideas I tried but all of them also failed to produce the correct answer. Any help would be appreciated. Thanks guys (I apologize if I messed up something it's my first time!)

Last edited: Aug 14, 2012
2. Aug 14, 2012

### Staff: Mentor

Re: Induced EMF in Rectangular Loop of Changing Area in Uniform/Constant Magnetic Fie

I don't think you can simplify it like that. Just write out the equation for the area A(t), then differentiate it for A'(t). Use that in the equations you show to find the induced EMF at t=4s.

3. Aug 14, 2012

### nickmanc86

Re: Induced EMF in Rectangular Loop of Changing Area in Uniform/Constant Magnetic Fie

Thank you so much that totally led me in the correct direction. I took A'(t)=L'*W+W'*L which I substituted in the equation for Induced EMF then plugged in a time of 4 and came to the correct answer of 3.6mV.

4. Aug 14, 2012

### Staff: Mentor

Re: Induced EMF in Rectangular Loop of Changing Area in Uniform/Constant Magnetic Fie

Good job!

Welcome to the PF, BTW.

5. Aug 14, 2012

### Staff: Mentor

Re: Induced EMF in Rectangular Loop of Changing Area in Uniform/Constant Magnetic Fie

Also, click "Quote" on your original post to see how I fixed the formating of your Latex...

6. Aug 14, 2012

### nickmanc86

Re: Induced EMF in Rectangular Loop of Changing Area in Uniform/Constant Magnetic Fie

Cool, Thank You

7. Apr 17, 2013

### iriver4

?

How did you get A'(t) ? Isn't the derivative of a constant zero ?

8. Apr 17, 2013

### Staff: Mentor

Welcome to the PF.

Please note that this thread has been dormant for almost a year. The OP has not posted since this thread, so most likely they are not around to answer your question...

9. Apr 17, 2013

### iriver4

Thank you

And would you be able to further explain how he got the problem ? I understand everything except how A'(t) is not equal to zero.

10. Apr 17, 2013

### Staff: Mentor

The area is not constant. The length and width are different, so there is a change in area, even though 2 sides are expanding while 2 sides are contracting. What do you get as an equation for the area as a function of time?

11. Apr 17, 2013

### iriver4

He posted that A'(t) = W'L + L'W
L = 60cm
W = 40cm

A'(t) = 0*60 + 0*40 = 0

Because the derivative of a constant is zero right ?

12. Apr 17, 2013

### Staff: Mentor

I'm not sure about what he posted, but the area is A(t) = L(t)*W(t) = (L+L'*t)(W-W'*t).

If you plug in the numbers from the problem statement, that area should be changing...

13. Apr 17, 2013

### iriver4

What are the values for L' and W' ? Im really sorry but I tried infinite ways and I cant get the correct answer

14. Apr 17, 2013

### Staff: Mentor

The problem statement says "rate of 20 mm/s"