Chapter 19: Ampere’s Law

19.1 Introduction

In this chapter, we will introduce Ampere’s law, which relates the magnetic field around a closed loop to the total electric current passing through the loop. Ampere’s law is an essential tool for calculating the magnetic fields generated by steady currents in wires and other conductive materials.

19.2 Ampere’s Law

Ampere’s law states that the line integral of the magnetic field along a closed loop, called the Amperian loop, is equal to the product of the permeability of free space and the total current passing through the loop:

The direction of the magnetic field follows the right-hand rule, where the thumb points in the direction of the current, and the fingers curl in the direction of the magnetic field.

19.3 Magnetic Field Due to a Straight Wire

Ampere’s law can be used to determine the magnetic field around a straight wire carrying a steady current . For a wire of infinite length, the magnetic field at a distance from the wire is given by:

19.4 Solenoids and Toroids

A solenoid is a helical coil of wire that generates a nearly uniform magnetic field inside when a current flows through it. Using Ampere’s law, the magnetic field inside a solenoid can be found:

where is the number of turns per unit length, and is the current flowing through the wire.

A toroid is a solenoid bent into a circular shape, with the ends connected to form a closed loop. The magnetic field inside a toroid is given by:

where is the total number of turns and is the distance from the center of the toroid.

19.5 Ampere’s Law and Maxwell’s Equations

Ampere’s law is one of the four fundamental Maxwell’s equations that describe the behavior of electric and magnetic fields. The other three equations are Gauss’s law for electricity, Gauss’s law for magnetism, and Faraday’s law of electromagnetic induction. Together, these equations form the foundation of classical electromagnetism.

Chapter Summary

In this chapter, we discussed Ampere’s law, which relates the magnetic field around a closed loop to the total current passing through the loop. We also covered the applications of Ampere’s law for calculating magnetic fields around straight wires, solenoids, and toroids. Ampere’s law is an essential principle in electromagnetism and a cornerstone of the broader set of Maxwell’s equations.

Continue to Chapter 20: Faraday’s Law, Lenz’s Law

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