Difference between a solenoid and a stack of loops

[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.

 

[mfb]

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.

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?

a and l are different things.

 

[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.

 

[mfb]

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.

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 want a serious answer

I posted one.

Thank you for your help.

You're welcome.

 

[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.