# Biot-Savart's Law for cylindrical conductor

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1. Apr 29, 2016

### Sho Kano

1. The problem statement, all variables and given/known data
The figure shows a cross section across a diameter of a long cylindrical conductor of radius a = 2.92 cm carrying uniform current 151 A. What is the magnitude of the current's magnetic field at the center of the conductor?

2. Relevant equations
Biot-Savart's Law

3. The attempt at a solution
B for circular loop = ui/2r
B = ui/2a
B = 0.0032 T
Which is the wrong answer, what happened?

2. Apr 29, 2016

### TSny

This formula is for the magnetic field at the center of a circular loop of current. But in your problem the current is not flowing in a circular loop. It is flowing in a long, straight, cylindrical conductor.

3. Apr 30, 2016

### Sho Kano

Is this still do-able with the Biot Savart law?

4. Apr 30, 2016

### TSny

Yes, you can easily find B at the central axis by using the Biot-Savart law and symmetry.

However, if you want to find B for an arbitrary value of r in the picture, then it would be easier to use another law.

5. Apr 30, 2016

### Sho Kano

In this case, I think the current is flowing through the center. Then that means there is no field at the center right?

6. Apr 30, 2016

### TSny

I'm not understanding this argument. Can you elaborate? The current is flowing at all points of the cylinder, not just along the central axis.

7. Apr 30, 2016

### Sho Kano

My bad, then can you show me how to do this problem? We have only learned so far the law for wires.

8. Apr 30, 2016

### TSny

The problem only asks for the B field at the center axis of the cylinder. For this, you can just use symmetry arguments. Think of the total current distribution as made up of a lot of long, parallel, straight filaments of current. Each filament is like a thin, straight wire carrying current. Use what you know about the direction of the B field due to a long, straight wire.

9. Apr 30, 2016

### Sho Kano

For parallel wires carrying the same current, there will be no net magnetic field at a point between them. So can I generalize this to the situation here?

10. Apr 30, 2016

### TSny

Yes. Good.

11. Apr 30, 2016

### Sho Kano

Now the problem asks for the field at radial distance 1 cm. How can I use symmetry for this?

12. Apr 30, 2016

### TSny

Now you need to do some math! The law of choice would be Ampere's law, not the Biot-Savart law. Symmetry will still be important.

13. Apr 30, 2016

### Sho Kano

Edit: I will come back to this

Last edited: Apr 30, 2016
14. Apr 30, 2016

### Sho Kano

Just watched a quickie on Ampere's Law. So I'm getting this:
ui = ∫B⋅dl
ui = B∫dl from 0 to 2πr
ui/2πr = B = 0.0030 T

Edit: I'm missing a current ratio?

15. Apr 30, 2016

### Sho Kano

Current per Area = 151 / (.0292^2 * pi) = 5.637 x 10^4 A/m^2
Current for Area of 1 cm radius = 5.637 x 10^4 A/m^2 * pi(.01^2)
= 3.141592653589794e-04

Use this for ampere's law gets:
3.5420e-04 T

16. Apr 30, 2016

### TSny

Looks good.

17. Apr 30, 2016

### Sho Kano

Now at, the wire's surface, It encloses the total current, at the radius of the wire. From the Ampere's Law, I get 0.0010T which is the wrong answer?

18. Apr 30, 2016

### TSny

I think that's the right answer. (Unless you need to get the number of significant figures correct also.)

19. Apr 30, 2016

### Sho Kano

I was marked wrong, I'll get back after asking the professor