The average EMF in a coil rotating in a magnetic field

[Asmaa Mohammad]

Homework Statement

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Homework Equations

The Attempt at a Solution

Solution for Q1:

Solution for Q2:

Solution for Q3:

Are they correct?

 

[cnh1995]

You have taken the peak emf to be 0.04V while the question says it is 0.4V.

Also, how did you calculate the average of emf for 1/4 revolution?

Solution for Q2 looks fine, but you should take the peak emf to be 0.4V.

For Q3: The frequency of emf is 1Hz. After 3 seconds, how many degrees would elapse? What is the emf at the beginning of this 3 second interval?

 

[Asmaa Mohammad]

You have taken the peak emf to be 0.04V while the question says it is 0.4V.

Ah, yes, This electronic edition of this exam is different from the one I have, luckily, the peak of the voltage is the only difference, I have just noticed.

Also, how did you calculate the average of emf for 1/4 revolution?

As this:
The change in the flux dΦ = BA - 0 (BA is the maximum flux linkage when the coil is perpendicular to the field and 0 is flux linkage when the coil is parallel to the field lines) So the change would be (BA - 0).
The change in time dt = T/4 (Where T is period), and since T=1/f -----> dt = 1/4f
The average emf = -N dΦ/dt = -NBA* 4f

For Q3: The frequency of emf is 1Hz. After 3 seconds, how many degrees would elapse? What is the emf at the beginning of this 3 second interval?

The coil starts its motion from the position in which it is parallel to the field lines, I think after 3 seconds it will come back to its original position, where it has a maximum induced emf. But I can't prove this mathematically.

 

[cnh1995]

The coil starts its motion from the position in which it is parallel to the field lines, I think after 3 seconds it will come back to its original position, where it has a maximum induced emf. But I can't prove this mathematically.

Right.
You know the initial emf, which is maximum.
So what is the general equation of emf in terms of maximum emf and θ, where θ is the angle between normal of the plane and the magnetic field?

The change in the flux dΦ = BA - 0 (BA is the maximum flux linkage when the coil is perpendicular to the field and 0 is flux linkage when the coil is parallel to the field lines) So the change would be (BA - 0).
The change in time dt = T/4 (Where T is period), and since T=1/f -----> dt = 1/4f
The average emf = -N dΦ/dt = -NBA* 4f

Right.

 

[Asmaa Mohammad]

So what is the general equation of emf in terms of maximum emf and θ, where θ is the angle between normal of the plane and the magnetic field?

It is:
emf = -NBAω sinθ = (emf)max*sinθ

But if you mean that I should use the equation above, then when should I use this one:
emf = (emf)max sinθ
θ = 2π*f* t ----> Where π =180, f is frequency and t is the time.

 

[cnh1995]

It is:
emf = -NBAω sinθ = (emf)max*sinθ

Yes. Minus sign is optional.

emf = (emf)max sinθ

Isn't it same as above?

 

[Asmaa Mohammad]

Isn't it same as above?

I meant it will be like this:
emf = (emf)max*sin (2πft).
t is the time, f is frequency, and π here is 180 not 3.14.

But I have difficulty in understanding which time I should use here, In my textbook when it uses this equation the coil starts its motion from the position in which it is perpendicular to the field lines, for example:
" a coil starts its motion from the position it is perpendicular to the field lines, the maximum induced emf in it, for instance, is 100V, and its frequency, for instance, is 50 Hz. Calculate the instantaneous emf after 1/200 seconds."

In problems like this one, my book uses the equation: emf = (emf)max*sin (2π*f*t) -----> where t is 1/200 seconds.

Have you understood my point of view??

 

[cnh1995]

So you are confused between when to take pi=180 and pi=3.14?

 

[Asmaa Mohammad]

So you are confused between when to take pi=180 and pi=3.14?

No, in this equation we should use pi = 180, and I have already used this equation in solution for Q3, look again!
θ = 2πft = 2*180* 1 * 3 = 1080
sin 1080 = 0
Then, emf = (emf)max* sin1080 = 0.4* 0 = 0
So, we will end up with zero induced emf and that's what will not actually happen ( I think that we both agree that after 3 seconds the induced emf will be maximum).

What confuses me is why this form of equation gives correct answer when the coil starts its motion from the position in which it is perpendicular to the field lines, but it doesn't work when the coil starts its motion from the parallel position??

 

[cnh1995]

E=E_maxsinθ
where θ is the angle between normal of the plane and the magnetic field.
At t=0, θ=90°. Therefore θ=90°-ωt and hence, θ=90°-2πft.
Now use this θ in the above emf equation ans you'll see that E=E_maxsin(90°-2πft)=E_maxcos(2πft).

If you want to use θ=ωt=2πft, you need to take θ as the angle between plane of the coil and the magnetic field. The emf equation will now become,
E=E_maxcos(2πft).

You'll end up with the same equation no matter which angle you choose.

 

[Asmaa Mohammad]

At t=0, θ=90°. Therefore θ=90°-ωt and hence, θ=90°-2πft.

So, when time passes, (θ = 90 -ωt) in the case we have the problem here,
but for the case in the problem mentioned in #7, it will be like this:
at t=0, θ=0. Therefore θ = 2πft.
right?

 

[cnh1995]

a coil starts its motion from the position it is perpendicular to the field lines, the maximum induced emf in it

This is not correct in your #7. The emf in this case will ve zero.

The expression for θ does not depend on the starting position of the coil. It depends on what you call as θ: the angle between plane of the coil and magnetic field or the angle between normal to the plane of the coil and the magnetic field.

So, when time passes, (θ = 90 -ωt) in the case we have the problem here,
but for the case in the problem mentioned in #7, it will be like this:
at t=0, θ=0.

You can use θ=90-ωt in #7 as well. At t=0, θ=90°, which is correct since θ is the angle between normal and the magnetic field, which is 90° when the coil is parallel to the field.

 

[rowan mamdouh]

You have taken the peak emf to be 0.04V while the question says it is 0.4V.

Also, how did you calculate the average of emf for 1/4 revolution?

Solution for Q2 looks fine, but you should take the peak emf to be 0.4V.

For Q3: The frequency of emf is 1Hz. After 3 seconds, how many degrees would elapse? What is the emf at the beginning of this 3 second interval?

You have taken the peak emf to be 0.04V while the question says it is 0.4V.

Also, how did you calculate the average of emf for 1/4 revolution?

Solution for Q2 looks fine, but you should take the peak emf to be 0.4V.

For Q3: The frequency of emf is 1Hz. After 3 seconds, how many degrees would elapse? What is the emf at the beginning of this 3 second interval?

why we don'T SUBSTITUE the negative sign in the solution ??