Magnetic field produced by coil and solenoid

[mikelepore]

Can someone please tell me why the formula for B at the center of a coil and the formula for B inside a solenoid give widely different answers in this problem?

From Serway and Vuille, College Physics: "A solenoid with 500 turns, 0.10 m long, carrying a current of 4.0 A and with a radius of 10^-2 m will have what strength magnetic field at its center?" -- Answer provided by the publisher: 250 X 10^-4 T

My comparison of the two formulas ...

Method 1
Using the formula for the magnetic field at the center of a coil that has N turns:

B = N mu0 I / 2 R = (500 turns) (4 pi X 10^-7 T m/A) (4.0 A) / 2 (10^-2 m)
= 0.12566 T = 0.126 T
(About five times greater than the publisher's answer)

Method 2
Using the solenoid formula, which textbooks describe as becoming more accurate as the windings become more closely packed, and as the length of the solenoid becomes much larger than its diameter, and if those assumptions are met then the formula expresses the nearly uniform field everywhere inside the solenoid, not only at the center:

turns per unit length = n = N/L = 500 turns / 0.10 m = 5000 turns per meter
B = mu0 n I = (4 pi X 10^-7 T m/A) (5000 turns per meter) (4.0 A)
= 0.025133 T = 0.025 T
(Agrees with the publisher's answer.)

Does the "coil" formula cease to become applicable just because the additional assumptions for using the "solenoid" formula have been provided? If so, why?

Thank you.

 

[Mindscrape]

No, method 1 won't work because you aren't simply merging the wires together, you are stacking them. Because you are stacking the loops by the time you get to the last loop you are no longer in the center of the loop and the equation you're trying to use is very wrong at this point. They won't just add one by one (for multiplication by N) as you suggested.

You could get the actual result, method 2 is only an approximation, if you were to take the equation for a loop and integrate. I seem to remember the integral is not so easy to do by hand, and you may not even know about integration, so I'm not going to go into it. The book described the conditions for using the infinite length solenoid approximation, which are mostly met by this problem.

 

[Mindscrape]

My bad, my computer froze, and I accidentally double posted.