# Solenoid vs coil of wire

This may sound pretty basic, but:

I'm studying for my AP Physics exam, and I am confused.

Apparently, the formula for the magnetic field in the center of a loop of wire with $n$ turns is:

$B=\frac{\mu_0NI}{2R}$.

However, the formula for the magnetic field in the center of a solenoid of length $L$ with $n$ turns is $B=\frac{\mu_0NI}{L}$.

Those two formulas are clearly not the same, but I do not see how a loop of wire with many turns is different from a solenoid. Both consist of current-carrying wire tightly wrapped into many coils, right? (Well, I get that the loop of wire is not necessarily tightly wrapped, but I am not aware of anything that says it cannot be.)

What's going on?

Thanks.

• You can think about the solenoid as tightly packed circular rings arranged over a large length (compared to the radius). At the center of the solenoid, the magnetic field is not due to the single coil at the center but due to the entire circular rings. According to each such rings, the center point of the solenoid is at different distances. It is not the center of each circualr rings. That's why the two formulas are different. – UKH Apr 27 '16 at 3:14

The solenoid is shaped like a cylinder with length $L$.
while the loop of wire with $n$ turns is essentially still in a plane and so is shaped like a circle of radius $R$
Note that for a solenoid, with $L$ 'large enough', the magnetic field is essentially uniform inside while, for the $n$ turn loop, the magnetic field varies with the distance from the center.