# Difference between a solenoid and a stack of loops

1. May 1, 2013

### warrenchu000

The B field at the center of a stack of N circular loops each carrying current I is
B = mu0 * N * I / (2a)
where a is the radius of the loop, derived using Biot-Savart law.

The B field inside a solenoid is
B = mu0 * N * I / l
where l is the length of the solenoid, derived using Ampere's law.

Yet everywhere I searched it is always stated that a solenoid can be thought of as a stack of circular loops. Then why are the results different? Why can't I use Ampere's law on the stack of loops to get mu0 * N * l?

It is also stated that in a solenoid the B field at the ends of the solenoid is 1/2 of the B field inside the solenoid, or
B = mu0 * N * I / (2l)

Could that be it? That is, the B field calculated using Biot-Savart law for stack of loops is the same as that for a solenoid B field but ONLY at the ends of the solenoid?

I am citing Figures 28.14 and 28.24 in University Physics by Young and Freedman, 13th edition.

2. May 1, 2013

### Staff: Mentor

It looks like your "stack" has a negligible length.

The formula for the solenoid assumes an "infinite" length.

Different setups lead to different formulas with different results.

a and l are different things.

3. May 1, 2013

### warrenchu000

No. There is nothing about the equation for the B field of a stack of rings being valid only for a short length. The number of loops is N and can be as large as I want. Moreover, the solenoid does not have to be infinite for the equation to be valid.

Of course I recognize the quantity "a" in the loop equation is the radius of the loop and "l" in the solenoid equation is the length of the solenoid. I am trying to reconcile these 2 formulas.

B = mu0 * N * I / (2a) for a stack of loops
B = mu0 * N * I / l for a solenoid

I want a serious answer, not just an off-the-cuff answer. I have been researching this for quite some time and have not found any article that addresses this question. Thank you for your help.

4. May 1, 2013

### Staff: Mentor

Just check where these equations come from, and which assumptions were made to derive them. If the stack of rings is allowed to have a variable length, this length would have to appear in the formula.

In the same way, I recognize the formula for solenoids, and it uses the approximation that the solenoid is very long relative to its diameter.
I posted one.
You're welcome.

5. May 1, 2013

### warrenchu000

I believe I have the answer.

B = mu0 * N * I / l is for the INSIDE of a solenoid far from the ends.

B = mu0 * N * I / (2a) is for a stack of loops on the END POINT of the stack.

Ampere's law was used to find the B field in the solenoid where it is ASSUMED it is uniform.

Ampere's law cannot be used for the B field of a stack of loops at points OUTSIDE the stack because the path does not enclose any current.

Moreover the B field on the outside is ASSUMED to be zero when applying Ampere's law. That is not true at the ends of the solenoid.

So these 2 formulas are for 2 completely different regions.