[Physics] Solenoid vs coil of wire

Question

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.

Answer 1

I do not see how a loop of wire with many turns is different from a solenoid

The solenoid is shaped like a cylinder with length $L$.

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while the loop of wire with $n$ turns is essentially still in a plane and so is shaped like a circle of radius $R$

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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.