EMF in a loop; non-constant magnetic field

[AJKing]

Homework Statement

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.

Homework Equations

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

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

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

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?

 

[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.]

 

[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)

 

[AJKing]

AH!

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

Thanks

 

[Nickie]

I would first double-check the given information and equations to ensure they are accurate and applicable to the scenario. In this case, it is important to note that the magnetic field in the loop is non-constant and will vary with time due to the changing current in the straight wire. This means that the Biot-Savart law for a straight wire may not be the most appropriate equation to use.

Instead, I would suggest using Faraday's law of induction, which states that the EMF induced in a loop is equal to the negative rate of change of the magnetic flux through the loop. This can be written as:

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

where \xi is the EMF and \Phi is the magnetic flux. In this case, the magnetic flux through the loop can be calculated by integrating the magnetic field over the area of the loop. This would give us:

\Phi = \int \int B \cdot dA

Since the magnetic field is non-constant, we would need to use the given expression for the current in the straight wire to calculate the magnetic field at any given time. Then, we can plug this into the above equation and solve for the EMF at t=3.0s.

It is also important to note that the EMF induced in the loop will depend on the orientation of the loop with respect to the magnetic field. In this case, the loop is perpendicular to the direction of the magnetic field, so we can use the maximum value of the magnetic field to calculate the EMF.

Overall, I believe the key to finding the correct solution lies in using Faraday's law of induction and properly incorporating the non-constant magnetic field in the calculation.