Max EMF in a Model AC Generator Coil

[mydoghenry]

Homework Statement

In a model ac generator, a 500 turn rectangular coil, 8.0 cm by 20 cm, rotates at 120 rev/min in a uniform magnetic field of 0.60 T.

(a) What is the maximum emf induced in the coil?
(b) What is the instantaneous value of the emf in the coil at t = (∏/32)s? Assume that the emf is zero at t = 0.
(c) What is the smallest value of t for which the emf will have its maximum value?

Homework Equations

εmax=NBAω
ε=εmaxsinωt

The Attempt at a Solution

I easily understand part a and b (below):
(a)
Area=0.08x0.20=0.016 m2
120rev/min = 12.56 rad/s

εmax=NBAω
εmax=500(0.60)(0.016)ω
εmax=60.28V

(b)
ε=εmaxsinωt
ε=60.28sin(12.56∏/32)
ε=56.87V

(c)
Part c is where I am totally confused. We did this problem in class, so I will show you what my teacher came up with and explain why I don't understand.

ε=εmaxsinωt
ωt=∏/2 --> ε=εmax
12.56t=∏/2
t=∏/2(12.56)=0.125 s

I do not understand where/how my teacher came up with ωt=∏/2. Why are we dividing pi by 2? And how do we know this is equal to ωt? Are we just supposed to know this equation? Is it a common physics rule? I want to know the "why" behind it and not just memorize that equation so I can apply it to other situations. For instance, what if we're asked what the largest value of t for which the emf will have it's maximum value?" Then it wouldn't still be pi over 2?

I have a test Thursday and have been wracking my brain trying to figure out why or how he came up with part c. I would greatly appreciate any and all help. Thank you so much! :!)

 

[tiny-tim]

welcome to pf!

hi mydoghenry!

ε=εmaxsinωt
ωt=∏/2 --> ε=εmaxsinωt
…
I do not understand where/how my teacher came up with ωt=∏/2.

the maximum value of sin (= 1) is when the angle is π/2

 

[mydoghenry]

Thank you so very, very much. It makes sense now. I must have missed the day in class when we learned ∏/2 is the radian form of 90°.

Do you think there's any other way for that question to be asked? For instance, What is the smallest value of t for which the emf will have its minimum value? In this case would it be sin180 (aka sin∏) which I guess is zero. So, maybe there's only one way he'd ask this question.

 

[tiny-tim]

So, maybe there's only one way he'd ask this question.

yes …

he's really not trying to trick you! ​

 

[Nickie]

Hello,

Firstly, it is important to understand the concept behind the equation ωt = π/2. This equation represents the angle at which the maximum emf occurs in the coil. In other words, at this angle, the coil is perpendicular to the magnetic field, resulting in the maximum change in flux and therefore the maximum induced emf.

To understand this concept, let's take a look at the diagram below:


As the coil rotates in the magnetic field, it experiences a change in flux due to the varying angle between the coil and the magnetic field. At t = 0, the coil is parallel to the magnetic field, resulting in no change in flux and therefore no induced emf. As the coil rotates, it reaches a point where it is perpendicular to the magnetic field, resulting in the maximum change in flux and therefore the maximum induced emf. This is represented by the angle π/2 in the diagram.

To understand why we divide π by 2, we need to look at the definition of angular velocity, which is ω = Δθ/Δt. In this case, Δθ represents the angle the coil rotates and Δt represents the time taken for the coil to rotate that angle. At the angle π/2, the coil has rotated one-quarter of a revolution, which is equivalent to π/2 radians. Therefore, ωt = π/2 represents the time taken for the coil to rotate one-quarter of a revolution, which is when the maximum emf is induced.

In summary, the equation ωt = π/2 represents the angle at which the maximum emf occurs in the coil, which is when the coil is perpendicular to the magnetic field. This equation is derived from the definition of angular velocity and the concept of change in flux. It is important to understand this concept rather than just memorizing the equation.

I hope this helps clarify your confusion. Best of luck on your test!