A long solenoid that has 1000 turns uniformly distributed...

A long solenoid that has 1000 turns uniformly distributed over a length of 0.400 $\mathrm{m}$ produces a magnetic field of magnitude $1.00 \times 10^{-4} \mathrm{T}$ at its center. What current is required in the windings for that to occur?

Key Concepts

Magnetic Field in a Solenoid

In electromagnetism, the magnetic field inside a long solenoid is described by the equation B = ?? n I, where B is the magnetic field, ?? is the permeability of free space, n is the number of turns per unit length, and I is the current. This formula assumes the magnetic field inside is uniform and is derived from applying Ampere's law to the solenoid geometry.

Ampere’s Law

Ampere’s Law is a fundamental relationship in electromagnetism that relates the line integral of the magnetic field around a closed loop to the total current passing through the loop. It provides the basis for calculating the magnetic field inside a solenoid by selecting an appropriate rectangular loop that aligns with the field lines.

Turn Density

Turn density is defined as the number of windings of the solenoid per unit length. It plays a critical role in determining the strength of the magnetic field inside the solenoid, as a higher turn density results in a greater magnetic field for a fixed current according to the formula B = ?? n I.

Permeability of Free Space

The permeability of free space, represented by ??, is a constant that characterizes the magnetic properties of the vacuum. It appears in key electromagnetic formulas including that for the magnetic field inside a solenoid, linking the current and the number of turns per unit length to the resulting magnetic field.

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