What does the strength of the magnetic field inside a circular conducting coil depend on?

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A circular conducting coil is called a solenoid; the magnetic field within the center of the solenoid is approximately uniform, and by Ampere's Law, its strength B is proportional to the permeability of the surrounding material (e.g. air, vacuum, etc.) `mu ` , the current running through the coil I, and the number of turns of the coil N divided by the length of the solenoid L:

`B = mu I N/L`

For an infinitely long solenoid (of course only a theoretical concept, not something we could actually build), we write it instead in terms of a density of loops, n:

`B = mu_0 I n`

For real solenoids, this is only an approximation, but it's often a quite good one.

Thus, the magnetic field inside the coil will be stronger if there is more current (higher I), if the material is more permeable (higher `mu` ), or if the coil is more tightly wound (higher n or higher N/L).

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