Dimensional check of the solenoid field

Problem

Using the dimensions of $\mu_0$ (T·m/A), of current (A) and of length (m), check dimensionally that the formula for the internal field of a solenoid, $B = \mu_0\,i\,N/L$, correctly gives a unit of tesla. Explain why $N$ is dimensionless.

Solution

Substituting the units of measurement of the quantities (with $N$ a pure number): $[B] = \frac{[\mu_0]\,[i]}{[L]} = \frac{(\text{T}\cdot\text{m/A})\cdot\text{A}}{\text{m}}$ The amperes cancel and the metres cancel, leaving the tesla. The number of turns $N$ is simply a count (how many turns there are), so it is a pure dimensionless number and does not alter the units of the result. $\ev{[B]=\dfrac{(\text{T}\cdot\text{m/A})\cdot\text{A}}{\text{m}}=\text{T}}$

Links

Topics: Magnetism Concepts: Magnetic field · Ampère’s theorem Skills: Dimensional analysis