Ampere's Law

Introduction

Outline

• Ampere's Law: Magnetism around Current
  • Electrical Charges and External Magnetic Fields
  • The Magnitude of Magnetic Force
• Practice with the Concepts
• Discussion Points

Ampere's Law: Magnetism around Current

Not only do moving charges give rise to magnetic fields, they also respond to external magnetic fields by changing direction.

Electrical Charges and External Magnetic Fields

When current flows through a wire, it really is just a lot of electrical charges in motion. If charge q moves through a wire of length l in time t, we can define this rate of motion either in terms of the velocity of the charge, or in terms of current i: $$q\ast v=q\ast \frac{l}{t}=\frac{q}{t}l=I\ast l$$

Ampere discovered that two wires with current flowing in the same direction will exert a mutually attractive force:

In a loop of wire, the field lines bend around the wire, reinforcing the field in the center.

If we have a loop of wire, the field gets more complicated (the force is not represented in this diagram):

The Magnitude of Magnetic Force

The strength of the force exerted depends on the current in the wire and the length of the wire is our old friend the vector cross product, or $$\overrightarrow{F}=I\ast \overrightarrow{l}\times \overrightarrow{B}$$ where θ is the angle between l (the length of the wire) and B. Note that when l and B are parallel, there is no force on the wire from the magnetic field.

If we have a particle moving with velocity v, then the force is $$\overrightarrow{F}=q\overrightarrow{v}\times \overrightarrow{B}=\mathrm{qvB}\sin \theta$$

The two forms, then, are equivalent:

$$F=\mathrm{qvB}\sin \theta =IlB\sin \theta$$

We use the first form to describe the resulting force acting on free charges moving in space through an external magnetic field, and the second to describe an external magnetic force acting on current-bearing wires.

All practical applications of electromagnetic fields — motors, generators, and anything that detects intensity or changes in either field — depend on this relationship.

Practice with the Concepts

Discussion Points

• Suppose that you have three iron rods, two of which are magnetized but the third is not. How can you determine which two are magnets without using any additional objects?
• Two insulated long wires carrying equal currents I cross at right angles to each other. Describe the magnetic force one exerts on the other.

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