Chapter 30:  Sources of the Magnetic Field

 

Learning Goals

  1. Know how to apply the Boit-Savart law.
  2. Be able to calculate the magnetic force between parallel conductors.
  3. Understand Ampere's law, and how it can be used to calculate magnetic fields for solenoids and torroids.
  4. Understand magnetic flux and how it applies to Gauss's law for magnetism. 
  5. Understand displacement current.

 

Biot-Savart Law

A steady current I flowing in a wire will produce a magnetic field at some point P (referred to as the field point) given by

 

where m0 = 4p x 10-7 T m /A, and r points from the current element ds to the field point P. In principle, the Biot-Savart law may always be used to solve for B, but it is not always the easiest approach.

 

Magnetic Field due to a Long Straight Wire

The magnetic field due to a long straight wire carrying a current I has magnitude given by

 

where r is the radial distance from the wire. The direction is azimuthal; that is, tangent to a circle centered on the wire in a sense given by the right hand rule.

 

Force Between Long Parallel Wires

The force between two long, parallel wires of length L carrying currents I1 and I2 and separated by a distance d has magnitude

 

 The force is attractive if the currents are in the same direction and repulsive otherwise.

 

Ampere’s Law

Let C be any closed path in space and let I be the net steady current that flows through C. Then

 

Like Gauss’s Law, Ampere’s Law is useful mostly for cases with enough symmetry to make the line integration simple to solve.

 

Solenoids and Toroids

An infinitely long coil (a.k.a solenoid) with n turns/unit length contains a constant, uniform magnetic filed with magnitude

 

The direction is along the axis of the coil.

A toroid is coil that is wrapped in a circle. The magnetic field is azimuthal along the axis of the toroid with magnitude

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

 

Magnetic Flux

Magnetic flux is defined in a way that is completely similar to our previous definition of electric flux:

 

 

Gauss’s Law for Magnetism

Since magnetic monopoles (magnetic "charges") have never been observed, Gauss’s law for magnetism is always

 

This is one of Maxwell’s Equations.

 

Displacement Current

Displacement current is associated with a time-varying electric flux:

 

 

Generalized Form of Ampere’s Law

Including the displacement current in Ampere’s Law makes it a Maxwell Equation:

 

In words, this law states that a changing electric field produces a magnetic field.