# Calculate the magnetic flux density

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1. Oct 21, 2016

### moenste

1. The problem statement, all variables and given/known data
The magnetic flux density in the vicinity of a large air-cored electromagnet is determined by measuring the induced flow of charge in a small coil as the current in the electromagnet is switched on -- see the diagram below.

Calculate the magnetic flux density at the position of the small coil, due to the electromagnet from the following data: radius of small coil = 10 mm, resistance per unit length of wire of small coil = 0.10 Ω m-1, total flow of charge through the small coil due to switching on the electromagnet = 0.01 C.

2. The attempt at a solution
Q = B A N / R → B = Q R / A N.

Q = 0.01, R = 0.1 Ω, A = ?, N = 1 turn (assumed).

A = π r2 = π * (10 / 10 / 100)2 = 3.14 * 10-4 m2.

B = 0.01 * 0.1 / 3.14 * 10-4 * 1 = 3.18 T.

What's wrong?

2. Oct 21, 2016

### TSny

Note that R is not given. Only the resistance per unit length is given.

Don't assume N = 1.

3. Oct 22, 2016

### moenste

Yes, I was also unsure on these things.

Regarding R: resistance per unit length of wire, while we only have radius. How can we find length?

Regarding N: how can we find the number of turns on a coil?

4. Oct 22, 2016

### TSny

You don't know the value of N. You will just have to include N in your expressions without knowing its value. Cross fingers that it cancels out in the end.
Can you express R in terms of N and r and the resistance per unit length?

5. Oct 22, 2016

### moenste

No idea how to do it. Can't find any formula which includes resistance radius and turns.

6. Oct 22, 2016

### TSny

If you knew the total length of wire in the coil, how would you calculate the total resistance of the coil?

7. Oct 22, 2016

### moenste

I can only think of R = ρ L / A, but we don't know resistivity ρ, and that's complicating even more.

8. Oct 22, 2016

### TSny

The resistance per unit length of the wire is given. Suppose we let the symbol u stand for the resistance per unit length. So, you are given that u = 0.01 Ω m-1. How do you interpret this information? For example, what would be the resistance of 3 meters of length of the wire?

9. Oct 22, 2016

### moenste

I would say that 0.01 * 3 = 0.03 Ω. So a wire of 3 m length has a resistance of 0.03 Ω.

10. Oct 22, 2016

### TSny

Yes. So, if the total length of wire in the coil is L, how would you express the resistance of the coil in terms of L and u?

11. Oct 22, 2016

### moenste

R / L = 0.01 Ω m-1. R = 0.01 L?

12. Oct 22, 2016

### TSny

OK. R = Lu. Working with symbols without plugging in numbers is a good idea for this problem. So, try to work out an expression for B in terms of N, r, u, and Q.

Can you express the total length of wire, L, in terms of r and N? How much is the length of one turn?

13. Oct 22, 2016

### moenste

Q = B A N / R → B = Q R / A N

A is π r2.

B = Q R / π r2 N.

R = L u so B = Q L u / π r2 N.

14. Oct 22, 2016

### TSny

Very good. Can you express L in terms of N and r?

15. Oct 22, 2016

### moenste

Can't find anything on it. No magnetic field formula fits this. Maybe something like L = π (2 * r) N?

16. Oct 22, 2016

### TSny

OK. To understand this, note that the coil has N turns. Each turn can be thought of as a circle of radius r. So, what is the length of one turn in terms of r? Then, what is the length of N turns?

17. Oct 22, 2016

### moenste

So 1 N = 2 r?

18. Oct 22, 2016

### TSny

No. Each turn of the coil consists of a length of wire that essentially has the shape of a circle of radius r. How long is a circle of radius r? That is, what is the formula from geometry that gives you the "length of a circle"?

19. Oct 22, 2016

### moenste

Like this?

20. Oct 22, 2016

### TSny

The length of a circle is also called the "circumference" of the circle.