In this problem, use Ampère's law to show that the magnetic...

In this problem, use Ampère's law to show that the magnetic field inside a long solenoid is $B=\mu_{0} n I .$ Assume that the field inside the solenoid is uniform and parallel to the axis and that the field outside is zero. Choose a rectangular path for Ampère's law. (a) Write down $B_{\|} \Delta l$ for each of the four sides of the path, in terms of $B, a$ (the short side), and $b$ (the long side). (b) Sum these to form the circulation. (c) Now, to find the current cutting through the path: each loop carries the same current $I,$ and some number $N$ of loops cut through the path, so the total current is $N I .$ Rewrite $N$ in terms of the number of turns per unit length $(n)$ and the physical dimensions of the path. (d) Solve for $B$.

In this problem, use Ampère's law to show that the magnetic field inside a long solenoid is $B=\mu_{0} n I .$ Assume that
the field inside the solenoid is uniform and parallel to the axis and that the field outside is zero. Choose a rectangular path for Ampère's law. (a) Write down $B_{\|} \Delta l$ for each of the four sides of the path, in terms of $B, a$ (the short side), and $b$ (the long side).
(b) Sum these to form the circulation.
(c) Now, to find the current cutting through the path: each loop carries the same current $I,$ and some number $N$ of loops cut through the path, so the total current is $N I .$ Rewrite $N$ in terms of the number of turns per unit length $(n)$ and the physical dimensions of the path.
(d) Solve for $B$.

Key Concepts

Current Enclosure in Multi-turn Coils

In a solenoid or any multi-turn coil, the net current enclosed by an Amperian loop is the sum of the contributions from each individual turn. This is often expressed in terms of the number of turns per unit length. Recognizing and calculating this collective contribution is essential to connecting the physical dimensions of the coil—such as its turn density—to the resulting magnetic field when Ampère's law is applied.

Magnetic Field Distribution in a Solenoid

This concept addresses the way in which the magnetic field is generated and distributed inside and outside a solenoid. In an ideal long solenoid, the field inside is approximated as uniform and strong, while the field outside is almost zero. This idealization allows for the straightforward use of Ampère's law to analytically determine the relationship between the current in the solenoid and the magnetic field produced.

Solenoid

A solenoid is a coil of wire designed to generate a magnetic field when an electric current flows through it. In an ideal long solenoid, the magnetic field inside is uniform and directed along its axis, while the field outside is very weak or nearly zero, making it a standard example in the application of Ampère’s law.

Ampère’s Law

A fundamental law of electromagnetism that relates the circulation of the magnetic field around a closed path to the net current passing through the area enclosed by that path. It is mathematically represented as ? B · dl = ??I_enc, where B is the magnetic field, dl is a differential element along the loop, ?? is the permeability of free space, and I_enc is the total current enclosed. This law is especially useful in systems with high symmetry, allowing for straightforward calculations of B.

Amperian Loop

An Amperian loop is an imagined closed curve chosen to exploit the symmetry of a current configuration when applying Ampère’s law. By selecting a path that aligns with the uniform regions of the magnetic field and avoids unnecessary complexity (such as regions where the field is negligible), the integration becomes simpler, making it easier to compute the magnetic field.

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