I was reading through a book and came across the following question and explanation:
You are given two 200 meter strands of identical copper wire. With one strand you create a coil whose radius is 2 cm. With the second strand you create a 4 cm coil. Assuming the current is the same in both, which coil will have the greater B field down its axis?
Solution: It turns out that the magnetic field expression for a coil is µoin, where µo is a constant and n is the number of turns per unit length in the coil. You might expect that B would have something to do with the coil's radius. After all, a bigger radius would mean more distance between the wire in the coil and the axis. But a bigger radius also means more length of wire per loop for the current to pass through, and the two parameters counteract one another. In any case, the magnetic fields should be the same.
I have tried to figure out what they were saying in this solution, but I was not able to figure i out.
I will explain what I tried to do in my attempt
Since a coil of wire is basically a solenoid, what we have is ampere's law and its usage in deriving an equation for a solenoid.
The strength of a magnetic field down a solenoid is:
B = mu0 * (Number of loops total / total solenoid LENGTH) * length of solenoid section * I
The Attempt at a Solution
Simplifying the equation, I examine the entire length of the solenoid, so I have
B = mu0 * Number of total loops * length of ENTIRE solenoid * current
Now, since I have one coil of 2 cm and another coil of 4cm, I will naturally have a greater number of loops "N" for the coil of 2 cm radius, so the B must be larger for that, is it not?
Then could someone please explain how I am mistaken and how I might recognize the mistake.
Thanks in advance for your input.