Calculate the magnetic flux density

[moenste]

Homework Statement

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.

Answer: 0.2 T.

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?

 

[TSny]

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

Don't assume N = 1.

 

[moenste]

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

Don't assume N = 1.

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?

 

[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?

 

[moenste]

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?

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

 

[TSny]

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

 

[moenste]

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

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

 

[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?

 

[moenste]

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?

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

 

[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?

 

[moenste]

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?

R / L = 0.01 Ω m^-1. R = 0.01 L?

 

[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?

 

[moenste]

So, try to work out an expression for B in terms of N, r, u, and Q.

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.

 

[TSny]

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

 

[moenste]

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

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

 

[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?

 

[moenste]

OK. To understand this, note that the coil has N turns. Each turn can be thought of as a circle of radius r.

So 1 N = 2 r?

 

[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"?

 

[moenste]

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"?

Like http://crackssc.com/sites/default/files/arclength.png?

 

[TSny]

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

 

[moenste]

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

2 π r?

 

[TSny]

Yes. So, how would you express the total length of wire in the coil in terms of N and r?

 

[moenste]

Yes. So, how would you express the total length of wire in the coil in terms of N and r?

R = L u so B = Q L u / π r² N.
L = 2 π r

B = Q 2 π r u / π r² N?

 

[TSny]

Since you want the resistance of the coil, L needs to be the length of wire in the coil. 2 π r is only the length of one turn of the coil.

 

[moenste]

Since you want the resistance of the coil, L needs to be the length of wire in the coil. 2 π r is only the length of one turn of the coil.

B = Q L u / π r² 2 π r, where N = 2 π r?

So b = Q L u / 2 π² r³.

But still no idea how to find the length of the wire. And also N = 2 π r is for only one turn. We don't know whether it has one or one hundred turns.

 

[TSny]

B = Q L u / π r²

Yes. Almost there. Just need the correct expression for L.

where N = 2 π r?

N is the number of "turns"; i.e., the number of "circles" making up the coil. 2 π r is the length of wire in one circle. N is just a number, 2 π r is a distance. So they can't be equal.

We don't know whether it has one or one hundred turns.

That's right. But you will see why you don't need to know the value of N.

So, if 2πr is the length of one circle and N is the number of circles, how can you write an expression for the total length L in terms of 2πr and N?

 

[moenste]

So, if 2πr is the length of one circle and N is the number of circles, how can you write an expression for the total length L in terms of 2πr and N?

I don't see any relationship between ten coils which have X length of their circles to correlate with the length of the whole coil. If we imagine our hand as a coil and four fingers are four turns the only way to find the length is 4 * diameter of each finger. Or 4 * (2 r).

 

[TSny]

If the coil had only one turn, what would be the length of wire in the coil?

If the coil had two turns, what would be the length of wire in the coil?

 

[moenste]

If the coil had only one turn, what would be the length of wire in the coil?

If the coil had two turns, what would be the length of wire in the coil?

1 turn: diameter or 2 r.
2 turns: 2 * diameter or 2 * 2 r.

N * (2 r)?

B = Q N 2 r u / π r² N = Q 2 u / π r?

 

[TSny]

Keep in mind that L is the length of the wire in the coil (because you need the resistance R of the wire). L is not the length of the cylinder on which the wire is wound. If you had to make a coil of N turns by winding wire around a cylinder of radius r, what length of wire would you need?