20.6 Force between Two Parallel Wires The magnetic field produced at the position of wire 2 due to the current in wire 1 is: The force this field exerts.

20.6 Force between Two Parallel Wires

The magnetic field produced at the position of wire 2 due to the current in wire 1 is:

The force this field exerts on a length l2 of wire 2 is:

(20-7)

20.6 Force between Two Parallel Wires

Parallel currents attract; antiparallel currents repel.

Example 20-10The two wires of a 2.0 m long appliance cord are 3.0 mm apart and carry a current of 8.0 A dc. Calculate the force one wire exerts on the other.

€

F =μ0

2π

I1I2

dl2

F =(2.0x10-7 Tm/A)(8.0 A) 2(2.0 m)

(3.0x10-3 m)

F = 8.5x10-3 N

The currents are in opposite directions (one toward the appliance, the other away from it), so the force would be repulsive and tend to spread the wires apart.

Example 20-11A horizontal wire carries a current I1=80 A dc. A second parallel wire 20 cm below it must carry how much current I2 so that it doesn’t fall due to gravity? The lower wire has a mass of 0.12 g per meter of length.

€

F = mg = (0.12x10-3 kg/m)(9.8 m/s2) =1.18x10-3 N/m

F =μ0

2π

I1I2

dl→ I2 =

2πd

μ0I1

F

l

I2 =2π (0.20 m)(1.18x10-3 N/m)

(4πx10-7 Tm/A)(80 A)=15 A

20.7 Solenoids and Electromagnets

A solenoid is a long coil of wire. If it is tightly wrapped, the magnetic field in its interior is almost uniform:

(20-8)

20.7 Solenoids and Electromagnets

If a piece of iron is inserted in the solenoid, the magnetic field greatly increases. Such electromagnets have many practical applications.

20.8 Ampère’s Law

Ampère’s law relates the magnetic field around a closed loop to the total current flowing through the loop.

(20-9)

20.8 Ampère’s Law

Ampère’s law can be used to calculate the magnetic field in situations with a high degree of symmetry.

20.9 Torque on a Current Loop; Magnetic Moment

The forces on opposite sides of a current loop will be equal and opposite (if the field is uniform and the loop is symmetric), but there may be a torque.

The magnitude of the torque is given by:

(20-10)

The quantity NIA is called the magnetic dipole moment, M:

(20-11)

Example 20-12A circular coil of wire has a diameter of 20.0 cm ad contains 10 loops. The current in each loop is 3.00 A, and the coil is placed in a 2.00 T external magnetic field. Determine the maximum and minimum torque exerted on the coil by the field.

€

A = πr2 = π (0.100 m)2 = 3.14x10-2 m

Max torque means sinθ =1

τ = NIABsinθ = (10)(3.00 A)(3.14x10 -2 m)(2.00 T)(1) =1.88 Nm

Min torque means sinθ = 0

τ = NIABsinθ = 0

20.10 Applications: Galvanometers, Motors, Loudspeakers

A galvanometer takes advantage of the torque on a current loop to measure current.

20.10 Applications: Galvanometers, Motors, Loudspeakers

An electric motor also takes advantage of the torque on a current loop, to change electrical energy to mechanical energy.

20.10 Applications: Galvanometers, Motors, Loudspeakers

Loudspeakers use the principle that a magnet exerts a force on a current-carrying wire to convert electrical signals into mechanical vibrations, producing sound.

20.11 Mass Spectrometer

A mass spectrometer measures the masses of atoms. If a charged particle is moving through perpendicular electric and magnetic fields, there is a particular speed at which it will not be deflected:

20.11 Mass Spectrometer

All the atoms reaching the second magnetic field will have the same speed; their radius of curvature will depend on their mass.

Example 20-13Carbon atoms of atomic mass 12.0 u are found to be mixed with another, unknown, element. In a mass spectrometer with fixed B’, the carbon traverses a path of radius 22.4 cm and the unknown’s path has a 26.2 cm radius. What is the unknown element? Assume they have the same charge.

€

mx

mc

=qBB'rx /E

qBB'rc/E=

26.2 cm

22.4 cm=1.17

mx =1.17x12.0 u =14.0 u, which corresponds

to nitrogen

20.12 Ferromagnetism: Domains and Hysteresis

Ferromagnetic materials are those that can become strongly magnetized, such as iron and nickel.

These materials are made up of tiny regions called domains; the magnetic field in each domain is in a single direction.

20.12 Ferromagnetism: Domains and Hysteresis

When the material is unmagnetized, the domains are randomly oriented. They can be partially or fully aligned by placing the material in an external magnetic field.

20.12 Ferromagnetism: Domains and Hysteresis

A magnet, if undisturbed, will tend to retain its magnetism. It can be demagnetized by shock or heat.

The relationship between the external magnetic field and the internal field in a ferromagnet is not simple, as the magnetization can vary.

20.12 Ferromagnetism: Domains and Hysteresis

Starting with unmagnetized material and no magnetic field, the magnetic field can be increased, decreased, reversed, and the cycle repeated. The resulting plot of the total magnetic field within the ferromagnet is called a hysteresis curve.

Summary of Chapter 20

• Magnets have north and south poles

• Like poles repel, unlike attract

• Unit of magnetic field: tesla

• Electric currents produce magnetic fields

• A magnetic field exerts a force on an electric current:

Summary of Chapter 20

• A magnetic field exerts a force on a moving charge:

• Magnitude of the field of a long, straight current-carrying wire:

• Parallel currents attract; antiparallel currents repel

Summary of Chapter 20

• Magnetic field inside a solenoid:

• Ampère’s law:

• Torque on a current loop:

Homework - Ch. 20

• Questions #’s 2, 3, 4, 5, 6, 14, 15

• Problems #’s 3, 7, 11, 15, 21, 29, 37, 49, 51, 53, 55, 61, 65