Induced EMF

1. Jul 2, 2015

gracy

1. The problem statement, all variables and given/known data

A conducting square frame of side a and a long straight wire carrying current I and located in the same plane as shown in the figure. The frame moves to the right with a constant velocity v. The emf induced in the frame will be proportional to

2. Relevant equations

3. The attempt at a solution
My teacher solved this in following manner

EMF induced in one side =B1vl
EMF induced in the other side=B2vl
EMF in the frame=B1vl - B2vl
∝ B1 - B2
Since B1∝1/r.I don't understand from where the below formula is coming?

Last edited by a moderator: Apr 30, 2017
2. Jul 2, 2015

Staff: Mentor

r is the distance between the current-carrying wire and each side. Express that distance in terms of the given x and a.

3. Jul 2, 2015

gracy

It was about proportionality.But My teacher gave full formula for magnetic field is B2= μ0 I /2 pi(X+a/2)
B1=μ0 I/2 pi(X - a/2)
But μ0 I/2 pi r is the formula for magnetic field due to a wire which is infinitely long from both side which is not the case here so I know it doesn't matter while taking proportionality into account but I wan to clarify this .

4. Jul 2, 2015

EM_Guy

For clarity, only the frame of the square is conducting - right? We don't have a square conductor - right? Just a current loop that is the shape of a square?

5. Jul 2, 2015

Staff: Mentor

What do you mean "from both sides"? The formula is for the magnetic field from an infinitely long current-carrying wire, which is a reasonable approximation here. Only one wire is involved. It has nothing to do with the length of the side of the square, if that's what you mean.

6. Jul 2, 2015

gracy

In my book there are 2 separate cases given
one which involves a wire of infinite length from both sides

And the second which has a wire with one starting point and the other end going to infinity.

Last edited: Jul 2, 2015
7. Jul 2, 2015

gracy

But it is not mentioned in the question that wire is infinitely long.
sir,I am aware of this fact.

8. Jul 2, 2015

Staff: Mentor

Ah, I see what you mean.

Now, looking at the diagram you posted originally, which case do you think they mean?

9. Jul 2, 2015

Staff: Mentor

You are told that it is a long straight wire. So it is reasonable to assume you can treat it as infinitely long, for the purposes of this problem.

10. Jul 2, 2015

gracy

Don't we have any formula for magnetic field due to normal conventional wire we see in our daily life ?

11. Jul 2, 2015

EM_Guy

Finite-length wires are a pain in magnetics. Think about the integration involved (especially if you have to deal with asymmetries). Blah! The fields produced by infinitely-long current-carrying conductors behave very nicely mathematically.

Remember, you can look at this from the inertial frame of the magnetic field or from the inertial frame of the square loop.

I would also advise considering each segment of the square loop separately.

The resulting expression for the emf should make sense. Think about these quantities: $\mu$, $I$, $a$, $v$, and $x$. Intuitively, how would increasing or decreasing these quantities affect your emf?

12. Jul 2, 2015

rude man

If a problem says "a long wire" you should assume they mean "infinitely long".

13. Jul 2, 2015

rude man

Yes. It's called the Biot-Savart law.
For example, you could have a closed square loop carrying current. In theory you can use B-Savart to compute the mag field anywhere. In practice the math in this case becomes prohibitively difficult except for calculating B at the center of the square.

14. Jul 2, 2015

EM_Guy

I do not agree that you want to calculate B at the center of the square. There is no conductor there, and we are trying to find the emf in the circuit.

This problem is a great problem for studying the so-called motional electric field intensity.

Also, if you have never studied the special theory of relativity, once you master the concepts of this problem, then it would be a great time to read Einstein's paper - at least the first page of it.

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15. Jul 2, 2015

Staff: Mentor

First learn to solve the simple problems, then you can add more complexity by considering finite lengths of wire. (As rude man said, that would involve the Biot-Savart law and calculus.)

16. Jul 2, 2015

Staff: Mentor

He was not suggesting that for this problem. He was giving an example of the complications involved with applying Biot-Savart.

17. Jul 2, 2015

EM_Guy

I see.

18. Jul 3, 2015

andrevdh

The magnetic field at a distance R form the wire with a current I in it is
B = k I/R
where k is some constant. The magnetic field at the point where the two conductors are
depends on where they are during the motion, x- a/2 for the one on the left and x+a/2 for
the one on the right.

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19. Jul 3, 2015

andrevdh

Since I is the same for both B1 and B2 the formula becomes even simpler

B = k 1/R