# PHYS 2212 Module 8

8: Sources of Magnetic Fields

An external hard drive attached to a computer works by magnetically encoding information that can be stored or retrieved quickly. A key idea in the development of digital devices is the ability to produce and use magnetic fields in this way. (credit: modification of work by “Miss Karen”/Flickr)

In the preceding module, we saw that a moving charged particle produces a magnetic field. This connection between electricity and magnetism is exploited in electromagnetic devices, such as a computer hard drive. In fact, it is the underlying principle behind most of the technology in modern society, including telephones, television, computers, and the internet.

In this module, we examine how magnetic fields are created by arbitrary distributions of electric current, using the Biot-Savart law. Then we look at how current-carrying wires create magnetic fields and deduce the forces that arise between two current-carrying wires due to these magnetic fields. We also study the torques produced by the magnetic fields of current loops. We then generalize these results to an important law of electromagnetism, called Ampère’s law.

We examine some devices that produce magnetic fields from currents in geometries based on loops, known as solenoids and toroids. Finally, we look at how materials behave in magnetic fields and categorize materials based on their responses to magnetic fields.

So far we’ve talked about the force that is exerted by a magnetic field on a moving charge. But we haven’t talked about where that magnetic field comes from, aside from a brief discussion on permanent magnets. But the magnetic fields we interact with every day (even though you might not know you’re dealing with B-fields) come from electric currents. Just like electric charge is the source of E-fields, electric current is the source of B-fields.

Take a look at this quick video that shows the magnetic field lines that are produced by a current running through a long, straight wire:

We will begin this module with a discussion of the Biot-Savart Law, which we use to determine the magnitude and direction of a magnetic field produced by a current (or any moving charge). Then we will use the Biot-Savart Law to determine the magnetic field produced by a current flowing in a straight wire, a circular loop, and a coil (called a solenoid).

After this, we will look at a different law — Ampere’s Law — for an alternative way to determine the magnetic field produced by a current.

Here are a couple more videos that I really like. Take a look!

#### 8.1 The Biot-Savart Law

• Explain how to derive a magnetic field from an arbitrary current in a line segment
• Calculate magnetic field from the Biot-Savart law in specific geometries, such as a current in a line and a current in a circular arc

#### 8.2 Magnetic Field Due to a Thin Straight Wire

• Explain how the Biot-Savart law is used to determine the magnetic field due to a thin, straight wire.
• Determine the dependence of the magnetic field from a thin, straight wire based on the distance from it and the current flowing in the wire.
• Sketch the magnetic field created from a thin, straight wire by using the second right-hand rule.

#### 8.3 Magnetic Field of a Current Loop

• Explain how the Biot-Savart law is used to determine the magnetic field due to a current in a loop of wire at a point along a line perpendicular to the plane of the loop.
• Determine the magnetic field of an arc of current.

#### 8.4 Magnetic Force between Two Parallel Currents

• Explain how parallel wires carrying currents can attract or repel each other
• Define the ampere and describe how it is related to current-carrying wires
• Calculate the force of attraction or repulsion between two current-carrying wires

#### 8.5 Ampère’s Law

• Explain how Ampère’s law relates the magnetic field produced by a current to the value of the current
• Calculate the magnetic field from a long straight wire, either thin or thick, by Ampère’s law

#### 8.6 Solenoids and Toroids

• Establish a relationship for how the magnetic field of a solenoid varies with distance and current by using both the Biot-Savart law and Ampère’s law
• Establish a relationship for how the magnetic field of a toroid varies with distance and current by using Ampère’s law

#### 8.7 Magnetism in Matter

• Classify magnetic materials as paramagnetic, diamagnetic, or ferromagnetic, based on their response to a magnetic field
• Sketch how magnetic dipoles align with the magnetic field in each type of substance
• Define hysteresis and magnetic susceptibility, which determines the type of magnetic material