Law for magnetic field due to solenoid

[NooDota]

Homework Statement

Just a quick clarification about the law: B = 4pi * 10^-7 * n *I/LDoes n refer to total number of loops, or just the number of loops in a single layer?

Like, If I have a solenoid that's made up of 4 layers and 1000 total loops, do I plug in 1000 or 250?

Homework Equations

The Attempt at a Solution

 

[Hesch]

Just a quick clarification about the law: B = 4pi * 10^-7 * n *I/L

Let me rewrite it:

B = μ₀*H

It looks like an attempt to calculate the B-Field in the center of a solenoide. Thus:

By "n/L", turns per length is meant, so you could write it:

B_center = μ₀ * I * ( ΔN / ΔL ).

n = ΔN , L = ΔL ( if you understand what I mean )

 

[Hesch]

Your formula is made from the figure beneath with Amperes law:

 

[rude man]

Homework Statement

Just a quick clarification about the law: B = 4pi * 10^-7 * n *I/L
Does n refer to total number of loops, or just the number of loops in a single layer?
Like, If I have a solenoid that's made up of 4 layers and 1000 total loops, do I plug in 1000 or 250?

1000.
n refers to the total number of loops in that formula.
However, conventionally n is used to mean "loops per unit length"; in SI that would be loops per meter.
It doesn't matter whether loops are in 1st or 2nd or 3rd or 4th layer, or any other layers.

 

[NooDota]

Okay, thank you all.

 

[Hesch]

n refers to the total number of loops in that formula.
However, conventionally n is used to mean "loops per unit length";

I don't think you are right here. It is not just a convention, but a premise. The intension of the formula is to calculate the H or B field at a distinct location ( the center ) in the solenoide.

Of symmetrical reasons you can only find the H and B fields at the center: Amperes law just tells you what the mean values will be, following the circulation path. Now calling the lower left corner of the rectangle ( circulation path ) in the figure in #3, A, then clockwise the other corners B, C, D. If AB and CD are very close to each other, the H fields parallel to AB and CD will be zero, because the H-field will be perpendicular to AB and CD due to symmetry. Making BC and DA longer ( or not centered ), the H-field will no longer be perpendicular to AB and CD. The lengths of AB and CD are assumed infinite, so that the H-field along BC is zero.

Therefore the formula can only find the H-field at the center of the solenoid, and thus DA must be kept very small. That's why n = ΔN / ΔL, ( not N/L ).

I know that the result will be the same, no matter if you use n or N. I'm just speaking of the premises as for the formula, and that the H-field is calculated at the center of the solenoid.
http://h2physics.org/wp-content/uploads/2010/05/solenoid2.jpg

 

[rude man]

I don't think you are right here. It is not just a convention, but a premise. The intension of the formula is to calculate the H or B field at a distinct location ( the center ) in the solenoide.

Of symmetrical reasons you can only find the H and B fields at the center: Amperes law just tells you what the mean values will be, following the circulation path. Now calling the lower left corner of the rectangle ( circulation path ) in the figure in #3, A, then clockwise the other corners B, C, D. If AB and CD are very close to each other, the H fields parallel to AB and CD will be zero, because the H-field will be perpendicular to AB and CD due to symmetry. Making BC and DA longer ( or not centered ), the H-field will no longer be perpendicular to AB and CD. The lengths of AB and CD are assumed infinite, so that the H-field along BC is zero.

Therefore the formula can only find the H-field at the center of the solenoid, and thus DA must be kept very small. That's why n = ΔN / ΔL, ( not N/L ).

I know that the result will be the same, no matter if you use n or N. I'm just speaking of the premises as for the formula, and that the H-field is calculated at the center of the solenoid.
http://h2physics.org/wp-content/uploads/2010/05/solenoid2.jpg

As before, I am confused by your riposte, feeling it to be inapposite.

 

[NooDota]

Ahh, I don't know any of the stuff you're talking about. I've only studied magnetic fields in my school and they just gave us direct laws to calculate them. (Straight line, solenoid, and a circular circuit or whatever you call it)

 

[rude man]

Ahh, I don't know any of the stuff you're talking about. I've only studied magnetic fields in my school and they just gave us direct laws to calculate them. (Straight line, solenoid, and a circular circuit or whatever you call it)

I wouldn't worry about it! (BTW your 'circular circuit' is probably a one-turn loop or a toroid).

 

[Hesch]

I am confused by your riposte, feeling it to be inapposite.

Why?

It's important to understand, that the H-field is not constant along a solenoid ( the H-field has most strength at the center ). As for other locations in/nearby the solenoid, Biot-Savart law must be used.

I'm just specifying that, as it is not mentioned in post #1.

How come that you are confused about that? Why is it inapposite?

This is the "wrong" sketch:

I've picked out the sketch in #6.

 

[rude man]

Why?.

Because you did not address the OP's question (post #1). Th OP's formula is the prima facie formula for axial B in a finite-length solenoid. The main purpose of its derivation is to show the multifarious utility of Ampere's law, and is always accompanied by a statement that the so computed field is approximate. I don't think the OP wanted to go beyond that point. I pointed out that the formula does not discriminate among windings from different layers. Thus, my answer of "1000" which is all he asked for.

For a more accurate computation of the axial B field inside or outside the solenoid one invokes the Biot-Savart law, applied to each winding individually and then summed for all windings. The ensuing summation is more simply replaced by the appropriate integral. Theodoros.mihos has recently posted a site which shows one way of accomplishing this:

 

[rude man]

That's why n = ΔN / ΔL, ( not N/L ).
Same thing, unless the spacings are uneven!

 

[Hesch]

Same thing, unless the spacings are uneven!

Its the same result, but not the same "thing".

Of symmetrical reasons you can only find the H and B fields at the center: Amperes law just tells you what the mean values will be, following the circulation path.

I've written this of educational reasons, so that the OP ( what does that mean? ) won't have to remember the formula. The OP has just to remember the principle in where the formula has derived from. Having understood that, and if the OP can somehow place a symmetric circulation path through a toroid, the OP can see through how the formula must be as for a toroid. ( H = N * I / 2πr ). I think it's more constructive to explain the idea ( how to find the H-field in the center of a toroid ) in the formula, than to reply with a "1000".

My english is not excellent, but I've found help here:

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html

( Same idea ).