Calculating Induced EMF for a Changing Magnetic Field Loop

In summary, a rectangular wire loop with dimensions of 60 cm length and 40 cm width is placed in a perpendicular and uniform magnetic field of 0.5 T. At time t=0, the loop begins to increase in length at a rate of 20 mm/s while also decreasing in width at the same rate. The induced emf at time t=4.0 s can be found by differentiating the equation for area A(t), which is (L+L'*t)(W-W'*t), and plugging in the given values to get the correct answer of 3.6 mV.
  • #1
nickmanc86
10
0
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

Answer: C




2. [tex]\varepsilon = -N \Delta \Phi /\Delta t[/tex] and [tex]\Phi = B*A[/tex]



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 [itex]\Delta A/\Delta t[/itex] 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!)
 
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  • #2


nickmanc86 said:
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

Answer: C




2. [tex]\varepsilon = -N \Delta \Phi /\Delta t[/tex] and [tex]\Phi = B*A[/tex]



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 [itex]\Delta A/\Delta t[/itex] 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!)

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


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


Good job! :smile:

Welcome to the PF, BTW.
 
  • #5


Also, click "Quote" on your original post to see how I fixed the formating of your Latex...
 
  • #6


Cool, Thank You
 
  • #7
?

How did you get A'(t) ? Isn't the derivative of a constant zero ?
 
  • #8
iriver4 said:
How did you get A'(t) ? Isn't the derivative of a constant zero ?

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
berkeman said:
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...

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
iriver4 said:
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.

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
berkeman said:
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?

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
iriver4 said:
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 ?

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
berkeman said:
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...

What are the values for L' and W' ? I am really sorry but I tried infinite ways and I can't get the correct answer
 
  • #14
iriver4 said:
What are the values for L' and W' ? I am really sorry but I tried infinite ways and I can't get the correct answer

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

FAQ: Calculating Induced EMF for a Changing Magnetic Field Loop

1. How is induced EMF calculated for a changing magnetic field loop?

The induced EMF (electromotive force) in a changing magnetic field loop can be calculated using Faraday's Law, which states that the induced EMF is equal to the negative rate of change of magnetic flux through the loop. This can be expressed as E = -dΦ/dt, where E is the induced EMF, Φ is the magnetic flux, and t is time.

2. What factors affect the induced EMF in a changing magnetic field loop?

The induced EMF in a changing magnetic field loop is affected by the rate of change of the magnetic field, the number of turns in the loop, and the area of the loop. Additionally, the presence of a core material within the loop can also affect the induced EMF.

3. How does the direction of the induced current in a loop relate to the changing magnetic field?

According to Lenz's Law, the direction of the induced current in a loop will always be such that it opposes the change in the magnetic field that is causing it. This means that if the magnetic field is increasing, the induced current will flow in a direction that creates a magnetic field that opposes the increase.

4. Can the induced EMF be negative in a changing magnetic field loop?

Yes, the induced EMF can be negative in a changing magnetic field loop. This would occur when the rate of change of the magnetic field is decreasing, causing the induced current to flow in a direction that creates a magnetic field that opposes the decrease.

5. How can the induced EMF in a changing magnetic field loop be increased?

The induced EMF in a changing magnetic field loop can be increased by increasing the rate of change of the magnetic field, increasing the number of turns in the loop, or increasing the area of the loop. Additionally, using a core material with a high magnetic permeability can also increase the induced EMF.

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