# Induced current in a metal ring

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1. Feb 6, 2017

### LogarithmLuke

1. The problem statement, all variables and given/known data
A metal ring with a radius of 10cm and resistance 0.1Ω passes through a delimited magnetic field, look at the photo posted below. The ring has a velocity of 5m/s, and the magnetic field has the field strength 0.4T. When the ring passes through the magnetic field, at some points an electrical current is induced in the ring.

Decide the direction and value of the current as the ring passes through.

2. Relevant equations

3. The attempt at a solution
I tried to find the induced voltage at spot number two, by using the formula: ε=(-π*0.1m*0.4t)/x
I wasnt really able to figure out the time, since no distances except the radius of the ring are given.

Last edited by a moderator: Apr 19, 2017
2. Feb 6, 2017

### TSny

Pictorial hint:

The yellow area represents the area over which there is flux at the present instant of time. During a small time interval Δt, an additional area represented by the blue strip will enter the region of B field.

3. Feb 6, 2017

### LogarithmLuke

What im having trouble with is finding the increase in area in respect to time.

4. Feb 6, 2017

### TSny

The increase in area is the area ΔA of the blue strip. Express the area of this rectangle in terms of the radius of the ring, the speed of the ring and Δt.

5. Feb 6, 2017

### Diegor

You have to find the function A(t) area under magnetic field at time t to calculate the flux. It looks that you have to do some integration to get A(t) but also after that some derivation (Faraday) so maybe they cancel each other, think about that.

6. Feb 7, 2017

### cnh1995

You can solve it using TSny's suggestion. It will involve some differentiation and transformation of variables.

Another approach is to use the effective length of the arc in the magnetic field. Effective length of the arc in the magnetic field will be simply the straight line distance between its two end points. Once you write that length as a function of given parameters, you can directly use the motional emf formula E=B*v*Leffective.

7. Feb 7, 2017

### TSny

No calculus or variable transformation is needed. Position "2" of the ring is special and finding the area of the blue rectangle is easy.

8. Feb 7, 2017

### cnh1995

I was talking about the general expression for emf E(t), not just for position 2.

Or is it possible to find E(t) just by using the speciality of position 2? I'll see if I can do that..
Thanks!

9. Feb 7, 2017

### TSny

OK, I see. I might have misinterpreted the problem. I was assuming that you only had to find the induced emf at the 5 specific positions shown.

10. Feb 7, 2017

### LogarithmLuke

The positions marked are basically general positions, so position 2 basically goes from where the ring is at in positon 2 in the picture, till it gets to position 3. Im excpected to find a min value, max value as well as an average value for the voltage in all positions.

11. Feb 7, 2017

### cnh1995

Can you express this effective length in terms of radius of the ring? Join the two ends of the arc in the magnetic field and express that length in terms of known quantities or variables.
Hint: Rectangular to polar conversion..

12. Feb 7, 2017

### TSny

OK. Sorry for the misinterpretation.

13. Feb 7, 2017

### malemdk

The area of the ring starts from zero and increases to maximum area , then constant for some time after that the area decreases to zero
time to increase the area from zero to max = 20/5000= 0.004s or 4ms ,

14. Feb 7, 2017

### LogarithmLuke

Should it not be 0.2m/5m/s=40ms ?

15. Feb 7, 2017

### cnh1995

The equation for emf comes out to be sinusoidal.

16. Feb 7, 2017

### malemdk

Yes, that correct

17. Feb 7, 2017

### malemdk

Since the ring intersects magnetic fluxes constantly for some time -depending on the width of fluxes -we cannot consider it as sinusoidal function

18. Feb 7, 2017

### cnh1995

I mean when the ring is entering the field region, the emf is sinusoidal. Once it is completely inside the field, there is no emf for some time. When it is leaving the region, there is again a sinusoidal emf.

19. Feb 7, 2017

### malemdk

When completely inside the field the emf will not be zero, since it is moving not stationary

20. Feb 8, 2017

### cnh1995

Though it is moving, the rate of change of flux through it is zero when it is completely inside the region. So there would be no emf in the ring once it is completely in the field.

Thinking in terms of motional emf, the effective length of the ring becomes zero when the ring is completely inside the field region, making the emf (BvLeffective) zero.