# EMF in a loop; non-constant magnetic field

1. Apr 6, 2013

### AJKing

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

Refer to Figure attached.

The current in the long straight wire is

i = (4.5A/s2)t2-(10A/s)t

Find the EMF in the square loop at t=3.0s.

2. Relevant equations

$\xi = -\frac{d \Phi}{dt}$

And Biot-Savart law for straight wires of infinite length:

$B = \frac{\mu_0 i}{2 \pi R}$

3. The attempt at a solution

Solution: 600 nV

I cannot recreate this result.

I consider two loops, one below the wire in the picture and one above.
I calculate their EMFS separately as:

$\xi = -\frac{A \mu_0}{2 \pi R} \frac{di}{dt}$

and find the difference between them.
This doesn't reveal 600 nV.

I try integrating with respect to R, and get mathematical gibberish (ln[0]).

I must be missing something fundamental in my setup - any suggestions?

#### Attached Files:

• ###### emfLOOP.jpg
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Last edited: Apr 6, 2013
2. Apr 6, 2013

### TSny

Think about the flux through the shaded areas in the figure.
[EDIT: This can help avoid dealing with r = 0. But I don't get 600 nV either. You will need to integrate.]
[Edit 2: OK, it does come out 600 nV.]

#### Attached Files:

• ###### emf loop.jpg
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Last edited: Apr 6, 2013
3. Apr 6, 2013

### AJKing

The change in flux for those areas cancel.

So, I consider the third region.

4cm away from the source.
8cm long, 16cm wide (not shown in figure)

$\xi = \frac{\mu_0 * 8cm * 16 cm * 17A/s}{2 \pi} \int^{12cm}_{4cm} \frac{1}{R}$ = 47 nV.

If I don't integrate, and instead just find the difference, I get closer, but it doesn't make sense to do that. ( = 725 nV)

4. Apr 6, 2013

### AJKing

AH!

You were right, it does come out to 600 nV.
Infinitesimal lengths :).

Thanks