Magnetic Field Calculations for Parallel Wires

[Kudo Shinichi]

Homework Statement

Four long copper wires are parallel to each other, their cross sections forming the corners of a square with sides a=20cm. A 20A current exists in each wire in the direction shown in the diagram.
http://i41.tinypic.com/2uiw6ck.jpg
a) What are the magnitude and direction of B at the center of the square?
b) What is the force per unit length acting on the lower left wire, in magnitude and direction?

The Attempt at a Solution

For both parts we need to find the magnetic field, so we can first use the right-hand rule to find the direction of the magnetic field for each point.
a) the magnetic field is turning in anticlockwise direction for the points that have the force pointing out, whereas the magnetic field is turning in clockwise direction for the points that have the force pointing in. both circular movements have the radius of 10cm(20cm/2).
We use the equation, B=(μo x I)/(2πr), to find the magnetic field for all points.
μo=4π x 10-7
I(for all points)=20
Therefore, the magnetic field for all points is 2x10-8
For the top side wire, the magnetic field is moving upward due to the interaction between the anticlockwise direction and the clockwise direction.
For the left side wire, the magnetic field is staying in the same location.
For the right side wire, the magnetic field is also staying in the same location.
For the bottom side wire, the magnetic field is also moving upward.

Magnetic field of B is the total magnetic field of all points, which is 8 x10-8 and it is moving upward.
b) since the question is just interest in the lower left wire, then I think that we just need to find out the magnetic field of the lower left point.
the magnetic field of the lower left point is turning in anticlockwise direction with radius of 10cm.
use the equation, B=(μo x I)/(2πr), to find the magnetic field for all points.
μo=4π x 10-7
I(for all points)=20
Therefore, the magnetic field for the lower left points is 2x10-8
Since there in no interaction with other point, so the magnetic field has anticlockwise direction.

I am not really sure whether I did the problem correctly or not. For part a all four points didn’t really touch the point B we are looking for, so I am not really sure whether add up the magnetic field for all four points would get the magnetic field for point B or not, but I think that I have the correct direction. For part b I am not really sure do I only use the lower left point to solve the problem.

Any comment or help would be great. Thank you very much.

 

[Doc Al]

For both parts we need to find the magnetic field, so we can first use the right-hand rule to find the direction of the magnetic field for each point.
a) the magnetic field is turning in anticlockwise direction for the points that have the force pointing out, whereas the magnetic field is turning in clockwise direction for the points that have the force pointing in.

OK, but you mean current, not force.

both circular movements have the radius of 10cm(20cm/2).

No, the distance to the center is half of the diagonal, not half the side.

We use the equation, B=(μo x I)/(2πr), to find the magnetic field for all points.
μo=4π x 10-7
I(for all points)=20
Therefore, the magnetic field for all points is 2x10-8

Correct the value of r.

For the top side wire, the magnetic field is moving upward due to the interaction between the anticlockwise direction and the clockwise direction.
For the left side wire, the magnetic field is staying in the same location.
For the right side wire, the magnetic field is also staying in the same location.
For the bottom side wire, the magnetic field is also moving upward.

This is hard to follow. Instead, draw arrows at the center representing the direction of the magnetic field from each wire. Then just add them up. (Some components will add; some will cancel.)

 

[Kudo Shinichi]

OK, but you mean current, not force.

No, the distance to the center is half of the diagonal, not half the side.

Correct the value of r.

This is hard to follow. Instead, draw arrows at the center representing the direction of the magnetic field from each wire. Then just add them up. (Some components will add; some will cancel.)

a)r=14.14
The magnetic field for B is 1.132x10^-6 and it is moving upward

I think that I have misread the question for part B, I just need to find out the magnetic field of the lower left point and use the force equation F=current times magnetic field times length to find out the force for the lower left wire, then use the right-hand rule and I found out that the force is pointing downward.

 

[Doc Al]

a)r=14.14

OK. (In cm.)

The magnetic field for B is 1.132x10^-6 and it is moving upward

How did you arrive at this? (I assume this is the total field? What did you get for the field from each wire?)

for part b do I just calculate the magnetic field for lower left point?

Yes, you'll need the field at that point to find the force.