Induced current in a metal ring

[cnh1995]

x = r cos(θ/2) where x is distance of center of ring from edge of field region and θ is subtended angle of effective length.

Ah..I see! I had written this equation earlier, don't know why I didn't contonue with that.
I'll work out the math later.

So L_effective=2rsinθ.
Which means motional emf=2Blvrsinθ.
This equation is sinusoidal w.r.t. the angle θ, but since θ doesn't vary linearly with time, the equation is not sinusoidal w.r.t. time. Is that correct?

 

[cnh1995]

I'll work out the math later.

Yes, the equation is indeed of an ellipse.
Thanks @TSny, @haruspex for catching the mistake.

 

[malemdk]

The area exposed to magnetic field is much more than simple Sine function

 

[haruspex]

The area exposed to magnetic field is much more than simple Sine function

Of course, but it's the rate of change of area that generates the emf.

 

[malemdk]

What I mean is that rate of change area is not simple function of Sine

 

[haruspex]

What I mean is that rate of change area is not simple function of Sine

"Function of Sine" is meaningless. Do you mean it is not the sine function of the subtended angle?
The rate of change of area is the linear speed multiplied by chord length. The chord length is 2r sin(θ)

 

[malemdk]

That's what try to mean,
But I have doubt, since the ring is closed loop how could emf develop?
Since no high or low potential point

 

[cnh1995]

That's what try to mean,
But I have doubt, since the ring is closed loop how could emf develop?
Since no high or low potential point

The induced electric field in the ring is non-conservative. Therefore, concept of potential is not applicable here.

Here, ∫_{closed loop}E⋅dl=dΦ/dt. In electrostatic case (circuits with battery) this integral for any loop is zero. The concept of potential is applicable in electrostatic field.

Here, the E field everywhere in the ring will be same (and non-zero), given by the above formula.

 

[malemdk]

You mean that the current keep on flowing?

Yes that makes sense

 

[cnh1995]

You mean that the current keep on flowing?

Yes.

 

[LogarithmLuke]

Has anyone found a way to solve it yet though? I am quite certain you don't need to use calculus to solve this, but I am sure there are multiple valid methods one can use.

 

[cnh1995]

Has anyone found a way to solve it yet though? I am quite certain you don't need to use calculus to solve this, but I am sure there are multiple valid methods one can use.

This problem needs trigonometry and some calculus (chain rule of differentiation).
The emf graph w.r.t. time is a part of an ellipse.
Read #23, #25, #28, #30 and #31..

 

[LogarithmLuke]

t

This problem needs trigonometry and some calculus (chain rule of differentiation).
The emf graph w.r.t. time is a part of an ellipse.
Read #23, #25, #28, #30 and #31..

That would probably work, but it seems like that would be more complicated than this really has to be. All the other problems in the textbook on induction have required far less complex methods than what you're describing. What i was thinking was to just find the average and max value for the emf in all the general positions. The problem doesen't really require us to represent all of the emf values induced with a graph, although I am sure its possible to solve the problem that way.

 

[cnh1995]

The problem doesen't really require us to represent all of the emf values induced with a graph, although I am sure its possible to solve the problem that way.

Then I guess they want you to find the voltage and current only at the 5 positions shown. Why else would they draw those 5 positions instead of just saying that the ring passes through the field with a constant velocity of 5m/s?

You can compute the emf at the given positions without calculus and any variable transformation. See TSny's pictorial hint in #2.

 

[TSny]

The problem doesen't really require us to represent all of the emf values induced with a graph, although I am sure its possible to solve the problem that way.

I think it would be helpful if you quoted the statement of the problem exactly as it was given to you. In the first post, you mention having to get the magnitude and direction of the induced current. Then in a later post, you mention having to get max, min, and average values of emf.

 

[LogarithmLuke]

I think it would be helpful if you quoted the statement of the problem exactly as it was given to you. In the first post, you mention having to get the magnitude and direction of the induced current. Then in a later post, you mention having to get max, min, and average values of emf.

I wrote the whole problem statement i my first post. The problem is that they did not specify whether they wanted the current as a function of the time or just spesific values. However in the answer section for the problems in the very back of the textbook they used a max value as well as an average value as an answer to this specific problem.

 

[LogarithmLuke]

So i was able to find the average value of the current for current for position 2. I found that the emf must be (((0.4T*pi*0.1m^2)/2)/20ms)/0.1Ω

This gave me 3.1A which apparently is the average current. However i do not really understand why that gave me the average current. Isnt the emf constant when part of the ring is in the magnetic field, then 0 when all of the ring is inside? Additionally, how do i go about finding the max value for the current?

 

[cnh1995]

Isnt the emf constant when part of the ring is in the magnetic field,

The emf is never constant (except when it is 0).

Are you asked to find the average emf from position 1 to position 5?

 

[LogarithmLuke]

The emf is never constant (except when it is 0).

Are you asked to find the average emf from position 1 to position 5?

Why isn't it constant when the speed is constant and the rate of change in area is constant?

The problem doesent specify, but based on the answers section were supposed to find the average and max values for each of the positions. However, positions 1,3 and 5 all have emf values of 0 (why is this?), so in reality only for positions 2 and 4.

Like i said the entire problem statement is in the first post, however the creators of the physics textbook were quite unclear in the way they formulated the problem. I am sure there is still a lot to learn through working with this though.

 

[cnh1995]

Why isn't it constant when the speed is constant and the rate of change in area is constant?

Area is not changing at a constant rate.

The emf graph w.r.t. time is a part of an ellipse.
Read #23, #25, #28, #30 and #31..

 

[haruspex]

That's what try to mean,

What is?
Are you still insisting that the induction is a sinusoidal function of time? Or do you now accept that the graph of it against time while crossing each wire is a half ellipse?

 

[Diegor]

Why isn't it constant when the speed is constant and the rate of change in area is constant?

The problem doesent specify, but based on the answers section were supposed to find the average and max values for each of the positions. However, positions 1,3 and 5 all have emf values of 0 (why is this?), so in reality only for positions 2 and 4.

Like i said the entire problem statement is in the first post, however the creators of the physics textbook were quite unclear in the way they formulated the problem. I am sure there is still a lot to learn through working with this though.

Did you try to get the increase of area (proportional to increase of flux) as a function of time? If you didn't see the picture maybe it will help you.