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Chapter 19: Magnetic Forces and Fields

- Magnetic Fields
- Magnetic Force on a Point Charge
- Motion of a Charged Particle in a Magnetic Field
- Crossed E and B fields
- Magnetic Forces on Current Carrying Wires
- Torque on a Current Loop
- Magnetic Field Due to a Current
- Ampère’s Law
- Magnetic Materials

§19.1 Magnetic Fields

All magnets have at least one north pole and one south pole.

Magnetic Dipole

Field lines emerge from north poles and enter through south poles.

Magnets exert forces on one another.

Opposite magnetic poles attract and like magnetic poles repel.

Magnetic field lines are closed loops. There is no (known!) source of magnetic field lines. (No magnetic monopoles)

If a magnet is broken in half you just end up with two magnets.

Near the surface of the Earth, the magnetic field is that of a dipole.

Note the orientation of the magnetic poles!

Away from the Earth, the magnetic field is distorted by the solar wind.

Evidence for magnetic pole reversals has been found on the ocean floor. The iron bearing minerals in the rock contain a record of the Earth’s magnetic field.

§19.2 Magnetic Force on a Point Charge

The magnetic force on a point charge is:

The unit of magnetic field (B) is the tesla (1T = 1 N/Am).

v

B

The magnitude of FB is:

where vsin is the component of the velocity perpendicular to the direction of the magnetic field. represents the angle between v and B.

Draw the vectors tail-to-tail to determine .

The direction of FB is found from the right-hand rule.

For a general cross product:

The right-hand rule is: using your right hand, point your fingers in the direction of A and curl them in the direction of B. Your thumb points in the direction of C.

y

x

v (west)

F (up)

Example (text problem 19.15): An electron moves with speed 2.0105 m/s in a 1.2 Tesla uniform magnetic field. At one instant, the electron is moving due west and experiences an upward magnetic force of 3.210-14 N. What is the direction of the magnetic field?

The angle can be either north of west OR north of east.

§19.3 Charged Particle Moving Perpendicular to a Uniform B-field

A positively charged particle has a velocity v (orange arrow) as shown. The magnetic field is into the page.

The magnetic force, at this instant, is shown in blue. In this region of space this positive charge will move CCW in a circular path.

Mass Spectrometer

B

A charged particle is shot into a region of known magnetic field.

Detector

V

Here,

Particles of different mass will travel different distances before striking the detector. (v, B, and q can be controlled.)

Other devices that use magnetic fields to bend particle paths are cyclotrons and synchrotrons.

Cyclotrons are used in the production of radioactive nuclei. For medical uses see the website of the Nuclear Energy Institute.

Synchrotrons are being tested for use in treating tumors.

§19.4 Motion of a Charged Particle in a Uniform B-field

If a charged particle has a component of its velocity perpendicular to B, then its path will be a circle. If it also a component of v parallel to B, then it will move forward as well. This resulting path is a helix.

§19.5 Crossed E and B Fields

If a charged particle enters a region of space with both electric and magnetic fields present, the force on the particle will be

B (into page)

E

Consider a region of space with crossed electric and magnetic fields.

Charge q>0 with velocity v

The value of the charge’s speed can be adjusted so that

The net force equal zero will occur when v=E/B.

This region of space (with crossed E and B fields) is called a “velocity selector”. It can be used as part of a mass spectrometer.

§19.6 Magnetic Force on a Current Carrying Wire

The force on a current carrying wire in an external magnetic field is

L is a vector that points in the direction of the current flow. Its magnitude is the length of the wire.

I= 1.0 A

B

Example (text problem 19.43): A 20.0 cm by 30.0 cm loop of wire carries 1.0 A of current clockwise.

(a) Find the magnetic force on each side of the loop if the magnetic field is 2.5 T to the left.

Left: F out of page

Top: no force

Right: F into page

Bottom: no force

Axis

B

L/2

L/2

§19.7 Torque on a Current LoopConsider a current carrying loop in a magnetic field. The net force on this loop is zero, but the net torque is not.

Force into page

Force out of page

The net torque on the current loop is:

N = number of turns of wire in the loop.

I = the current carried by the loop.

A = area of the loop.

B = the magnetic field strength.

= the angle between A and B.

The direction of A is defined with a right-hand rule. Curl the fingers of your right hand in the direction of the current flow around a loop and your thumb will point in the direction of A.

Because there is a torque on the current loop, it must have both a north and south pole. A current loop is a magnetic dipole. (Your thumb, using the above RHR, points from south to north.)

§19.8 Magnetic Field due to a Current

Moving charges (a current) create magnetic fields.

The magnetic field at a distance r from a long, straight wire carrying current I is

where 0 = 410-7 Tm/A is the permeability of free space.

The direction of the B-field lines is given by a right-hand rule. Point the thumb of your right hand in the direction of the current flow while wrapping your hand around the wire; your fingers will curl in the direction of the magnetic field lines.

A wire carries current I out of the page.

The B-field lines of this wire are CCW.

Note: The field (B) is tangent to the field lines.

1

I

2

d

I

Example (text problem 19.62): Two parallel wires in a horizontal plane carry currents I1 and I2 to the right. The wires each have a length L and are separated by a

distance d.

(a) What are the magnitude and direction of the B-field of wire 1 at the location of wire 2?

Into the page

Example continued:

(b) What are the magnitude and direction of the magnetic force on wire 2 due to wire 1?

F12 toward top of page (toward wire 1)

(c) What are the magnitude and direction of the B-field of wire 2 at the location of wire 1?

Out of the page

Example continued:

(d) What are the magnitude and direction of the magnetic force on wire 1 due to wire 2?

F21 toward bottom of page (toward wire 2)

(e) Do parallel currents attract or repel? They attract.

(f) Do antiparallel currents attract or repel? They repel.

The magnetic field of a current loop:

The strength of the B-field at the center of the (single) wire loop is:

The magnetic field of a solenoid:

A solenoid is a coil of wire that is wrapped in a cylindrical shape.

The field inside a solenoid is nearly uniform (if you stay away from the ends) and has a strength:

Where n=N/L is the number of turns of wire (N) per unit length (L) and I is the current in the wire.

§19.9 Ampère’s Law

Ampère’s Law relates the magnetic field on a path to the net current cutting through the path.

Example (text problem 19.65): A number of wires carry current into or out of the page as indicated.

(a) What is the net current though the interior of loop 1?

Example continued:

Assume currents into the page are negative and current out of the page are positive.

Loop 1 encloses currents -3I, +14I, and -6I. The net current is +5I or 5I out of the page.

(b) What is the net current though the interior of loop 2?

Loop 2 encloses currents -16I and +14I. The net current is -2I or 2I into the page.

Consider a wire carrying current into the page. Draw a closed path around the wire.

Here the B-field is tangent to the path everywhere (hence the choice of a circular path). The circulation is

Ampere’s Law is

where I is the net current that cuts through the circular path.

If the wire from the previous page carries a current I then the magnetic field at distance r from the wire is

§19.10 Magnetic Materials

Ferromagnetic materials have domains, regions in which its atomic dipoles are aligned, giving the region a strong dipole moment.

When the domains are oriented randomly there will be no net magnetization of the object.

When the domains are aligned, the material will have a net magnetization.

Summary

- Magnetic forces are felt only by moving charges
- Right-Hand Rules
- Magnetic Force on a Current Carrying Wire
- Torque on a Current Loop
- Magnetic Field ofa Current Carrying Wire (straight wire, wire loop, solenoid)
- Ampère’s Law

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