Derive an expression of magnetic field due to long current carrying solenoid?

To derive the expression for the magnetic field due to a long current-carrying solenoid, we can use Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the permeability of free space (μ₀) and the total current passing through the loop.

Consider a long solenoid with a length much greater than its diameter. The solenoid consists of a tightly wound coil of wire with N turns per unit length. Let's choose a closed loop parallel to the length of the solenoid and passing through the center of the coil. The loop has a length l and encloses a length l' of the solenoid.

According to Ampere's law, we have:

∮B·dl = μ₀I,

where B is the magnetic field, dl is an infinitesimal element along the loop, I is the current passing through the loop, and μ₀ is the permeability of free space.

The left-hand side of the equation represents the line integral of the magnetic field around the closed loop.

Since the loop is parallel to the length of the solenoid, the magnetic field B will be constant along the loop. Therefore, we can take B out of the integral:

B ∮dl = B l,

where B is the magnitude of the magnetic field.

The right-hand side of the equation is μ₀ times the total current passing through the loop. The total current passing through the loop is the product of the current per unit length (I₀) of the solenoid and the length l' of the solenoid enclosed by the loop:

I = I₀l'.

Combining these results, we have:

B l = μ₀ I₀ l',

Simplifying, we find:

B = μ₀ I₀,

This is the expression for the magnetic field inside a long current-carrying solenoid. The magnetic field magnitude B is directly proportional to the product of the permeability of free space (μ₀) and the current per unit length (I₀) of the solenoid.