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MyChapter 19

LectureOutline

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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

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§19.1 Magnetic Fields

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

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

Magnetic Dipole

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Magnets exert forces on one another.

Opposite magnetic poles attract and like magnetic poles repel.

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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.

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

Note the orientation of the magnetic poles!

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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.

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§19.2 Magnetic Force on a Point Charge

( )BvF ×= qB

The magnetic force on a point charge is:

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

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The magnitude of FB is: ( )θsinB vqBF =

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

θ

v

B

Draw the vectors tail-to-tail to determine θ.

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

The right-hand rule is: using your right hand, point your fingers in the direction of the velocity v and your thumb in the direction of the magnetic field B. The palm of your hand points in the direction of the force F.

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§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.

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r

vmqvB

maFF r

2

B

=

==∑

Applying Newton’s 2nd Law to the charge:

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Example: How long does it take an electron to complete one revolution if the radius of its path is r (see the figure on slide 11)?

The distance traveled by the electron during one revolution is d = 2r. The electron moves at constant speed so d = vT as well. The speed of the electron can be obtained using the result of the previous slide.

eB

m

meBrr

v

rT e

e

222=== Is the period of the

electron’s motion.

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Mass Spectrometer

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

mvqBrr

vmqvB

=

=

or

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

⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗

V

Detector

B

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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.

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§19.6 Magnetic Force on a Current Carrying Wire

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

( )BLF ×= I

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

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( )BLF ×= IThe magnitude of is

θsinILBF =

and its direction is given by the right-hand rule.

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Example (text problem 19.50): 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.

I = 1.0 A

B

Left: F out of page

Top: no force

Right: F into page

Bottom: no force

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( )( )( )N 50.0

90sinT 5.2m 20.0A 0.1

sin

=

°=

= θILBF

The magnitudes of the nonzero forces are:

(b) What is the net force on the loop?

0net =F

Example continued:

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§19.7 Torque on a Current Loop

Consider a current carrying loop in a magnetic field. The net force on this loop is zero, but the net torque is not.

B

Axis

L/2L/2

Force out of page

Force into page

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The net torque on the current loop is:

θτ sinNIAB=

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.

θ = the angle between A and B.

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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.)

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§19.8 Magnetic Field due to a Current

Moving charges (a current) create magnetic fields.

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

r

IB

μ2

0=

where μ0 = 4107 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.

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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.

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Example (text problem 19.72): 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?

I

I

d

1

2

d

IB

μ2

101 = Into the page

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

d

LIILBI

LBIF

μθ

2

sin

21012

1212

==

=

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?

d

IB

μ2

202 = Out of the page

Example continued:

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(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.

d

LIILBI

LBIF

μθ

2

sin

21021

2121

==

=

Example continued:

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The magnetic field of a current loop:

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

R

IB

20μ=

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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:

nIB 0μ=

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

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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 of a Current Carrying Wire (straight wire, wire loop, solenoid)