Magnetic Flux DensityMagnetic Flux Density

Magnetic Flux density:

ABA

• Magnetic flux lines Magnetic flux lines are continuous and are continuous and

closed.closed.• Direction is that of Direction is that of the B vector at any the B vector at any point.point.• Flux lines are Flux lines are NOTNOT in in direction of force but direction of force but ..

; = B BAA

; = B BAA

When area A is

perpendicular to flux:

When area A is perpendicular to

flux:

The unit of flux density is the The unit of flux density is the Weber per square Weber per square metermeter..

Calculating Flux Density Calculating Flux Density When Area is Not When Area is Not

PerpendicularPerpendicularThe flux penetrating The flux penetrating the area the area AA when the when the

normal vector normal vector nn makes an angle of makes an angle of with the with the B-fieldB-field is: is:

cosBA cosBA

The angle The angle is the complement of the angle a is the complement of the angle a that the plane of the area makes with the B field.that the plane of the area makes with the B field. (Cos (Cos = Sin = Sin

nA

B

Origin of Magnetic FieldsOrigin of Magnetic FieldsRecall that the strength of an Recall that the strength of an electric field Eelectric field E was defined as the electric force per unit was defined as the electric force per unit charge.charge.Since Since no isolated magnetic poleno isolated magnetic pole has ever has ever been foundbeen found, we can’t define the magnetic , we can’t define the magnetic field field B B in terms of the in terms of the magnetic force per magnetic force per unit north poleunit north pole..We will see instead that magnetic fields result from charges in motion—not from stationary charge or poles. This fact will be covered later.

We will see instead that magnetic fields result from charges in motion—not from stationary charge or poles. This fact will be covered later.

+E

+ B vv

Magnetic Force on Moving Magnetic Force on Moving ChargeCharge

N S

B

N

Imagine a tube Imagine a tube that projects that projects charge charge +q+q with with velocity velocity vv into into perpendicular perpendicular BB field.field.

Upward magnetic force F on charge moving in B

field.

vv

FF

Experiment shows:Experiment shows:

F qvBF qvB

Each of the following results in a greater Each of the following results in a greater magnetic magnetic force Fforce F: an increase in : an increase in velocityvelocity vv, , an increase in an increase in chargecharge qq, and a larger , and a larger magnetic field Bmagnetic field B..

Direction of Magnetic Direction of Magnetic ForceForce

B

vv

FF

N SN

The right hand The right hand rulerule::With a flat With a flat rightright hand, hand,

point point thumbthumb in in direction of velocity direction of velocity vv, , fingersfingers in direction in direction of of BB field. The flat field. The flat handhand pushes in the pushes in the direction of direction of force Fforce F..

The force is greatest when the velocity v is perpendicular to the B field. The deflection decreases to zero for parallel motion.

The force is greatest when the velocity v is perpendicular to the B field. The deflection decreases to zero for parallel motion.

B

vv

FF

Force and Angle of PathForce and Angle of Path

SNN

SNN

SNN

Deflection force Deflection force greatest when path greatest when path perpendicular to field. perpendicular to field. Least at parallel.Least at parallel.

sinF v sinF v

B

vv

FF

v sin v sin vv

Definition of B-fieldDefinition of B-fieldExperimental observations show the following:Experimental observations show the following:

sin or constantsin

FF qv

qv

sin or constant

sin

FF qv

qv

By choosing appropriate units for the By choosing appropriate units for the constant of proportionality, we can now constant of proportionality, we can now define the define the B-fieldB-field as: as:

or sinsin

FB F qvB

qv

or sin

sin

FB F qvB

qv

Magnetic

Field Intensity B:

A A magnetic field intensitymagnetic field intensity of one of one tesla (T)tesla (T) exists in a region of space where a charge of exists in a region of space where a charge of one coulombone coulomb (C)(C) moving at moving at 1 m/s1 m/s perpendicular to the B-field will experience a perpendicular to the B-field will experience a force of one force of one newton (N).newton (N).

A A magnetic field intensitymagnetic field intensity of one of one tesla (T)tesla (T) exists in a region of space where a charge of exists in a region of space where a charge of one coulombone coulomb (C)(C) moving at moving at 1 m/s1 m/s perpendicular to the B-field will experience a perpendicular to the B-field will experience a force of one force of one newton (N).newton (N).

Example 1.Example 1. A A 2-nC2-nC charge is projected with charge is projected with velocity velocity 5 x 105 x 1044 m/s m/s at an angle of at an angle of 303000 with with a a 3 mT3 mT magnetic field as shown. What magnetic field as shown. What are the magnitude and direction of the are the magnitude and direction of the resulting force? resulting force?

v sin v sin vv

B

vv

FFDraw a rough Draw a rough sketch.sketch.qq = 2 x 10 = 2 x 10-9-9 C C

vv = 5 x 10 = 5 x 1044 m/s m/s B B = 3 x = 3 x 1010-3-3 T T = 30= 3000

Using right-hand rule, the force is seen to beUsing right-hand rule, the force is seen to be upwardupward..

-9 4 -3 0sin (2 x 10 C)(5 x 10 m/s)(3 x 10 T)sin 30F qvB

Resultant Magnetic Force: F = 1.50 x 10-7 N, upward

Resultant Magnetic Force: F = 1.50 x 10-7 N, upward

B

Forces on Negative Forces on Negative ChargesCharges

Forces onForces on negative negative charges are opposite to charges are opposite to those on positive charges. The force on the those on positive charges. The force on the negative charge requires a negative charge requires a left-hand ruleleft-hand rule to to show show downwarddownward force force FF..

Forces onForces on negative negative charges are opposite to charges are opposite to those on positive charges. The force on the those on positive charges. The force on the negative charge requires a negative charge requires a left-hand ruleleft-hand rule to to show show downwarddownward force force FF..

N SN N SN

B

vv

FFRight-hand rule

for positive q

FF

B

vvLeft-hand rule for

negative q

Indicating Direction of B-Indicating Direction of B-fieldsfieldsOne way of indicating the directions of fields One way of indicating the directions of fields perpen-dicular to a plane is to use crosses perpen-dicular to a plane is to use crosses X X and dots and dots :

X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X X

A field directed into the paper is denoted by a cross “X” like the tail feathers of an arrow.

A field directed out of the paper is denoted by a dot “ ” like the front tip end of an arrow.

Practice With Directions:Practice With Directions:

X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X X

X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X X

What is the direction of the force F on the charge in each of the examples described

below?

What is the direction of the force F on the charge in each of the examples described

below?

-vv

-vv

+

vv

vv+

UpFF

LeftFF

FFRight

UpFF

negative negative qq

Crossed E and B FieldsCrossed E and B FieldsThe motion of charged particles, such as The motion of charged particles, such as electrons, can be controlled by combined electrons, can be controlled by combined electric and magnetic fields.electric and magnetic fields.

The motion of charged particles, such as The motion of charged particles, such as electrons, can be controlled by combined electrons, can be controlled by combined electric and magnetic fields.electric and magnetic fields.

x x x x x x

x x

+

-

e-

v

Note:Note: FFEE on on electron is electron is upwardupward and and opposite E-field.opposite E-field.But, But, FFBB on electron on electron is is downdown (left-hand (left-hand rule).rule).Zero deflection Zero deflection when when FFBB = F = FEE

B

vv

FFEE

E e--

B

vvFFBB

-

The Velocity SelectorThe Velocity SelectorThis device uses crossed fields to select only This device uses crossed fields to select only those velocities for which Fthose velocities for which FBB = F = FEE. (Verify . (Verify directions for +q)directions for +q)

This device uses crossed fields to select only This device uses crossed fields to select only those velocities for which Fthose velocities for which FBB = F = FEE. (Verify . (Verify directions for +q)directions for +q)

x x x x x x

x x

+

-

+qv

Source of +q

Velocity selector

When FWhen FBB = F = FE E ::

qvB qE

Ev

BE

vB

By adjusting the E and/or B-fields, a person can select only those ions with the desired

velocity.

By adjusting the E and/or B-fields, a person can select only those ions with the desired

velocity.

Example 2.Example 2. A lithium ion, A lithium ion, qq = +1.6 x 10 = +1.6 x 10--

1616 C C, is projected through a velocity , is projected through a velocity selector where selector where B = 20 mTB = 20 mT. The E-field is . The E-field is adjusted to select a velocity of adjusted to select a velocity of 1.5 x 101.5 x 1066 m/sm/s. What is the electric field E?. What is the electric field E?

x x x x x x

x x

+

-

+qv

Source of +q

VV

Ev

BE

vB

E = vBE = vB

E = E = (1.5 x 10(1.5 x 1066 m/s)(20 x 10 m/s)(20 x 10-3-3 T);T);

E = 3.00 x 104 V/m

E = 3.00 x 104 V/m

Circular Motion in B-fieldCircular Motion in B-fieldThe magnetic force F on a moving charge is always perpendicular to its velocity v. Thus, a charge moving in a B-field will experience a centripetal force.

The magnetic force F on a moving charge is always perpendicular to its velocity v. Thus, a charge moving in a B-field will experience a centripetal force.

X X X X X X X X X X

X X X X X X X X X X

X X X X X X X X X X

X X X X X X X X X X

X X X X X X X X X X

X X X X XX X X X X

++

+

+

Centripetal FCentripetal Fcc = = FFBB

RR

FFcc

2

; ;C B

mvF F qvB

R

2mvqvB

RC BF F

The radius of path is:

The radius of path is:

mvR

qBmv

RqB

Mass SpectrometerMass Spectrometer

+q

R

Ev

B

+-

x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x

x x x x

x x x x x x x x

Photographic plate

m1

m2

slit

Ions passed through Ions passed through a velocity selector a velocity selector at known velocity at known velocity emerge into a emerge into a magnetic field as magnetic field as shown. The radius shown. The radius is:is:

The mass is found by The mass is found by measuring the measuring the radius R:radius R:

mvR

qB

qBRm

vqBR

mv

2mvqvB

R

Example 3.Example 3. A Neon ion, A Neon ion, q = 1.6 x 10q = 1.6 x 10-19 -19 CC, , follows a path of radius follows a path of radius 7.28 cm7.28 cm. Upper . Upper and lower and lower B = 0.5 TB = 0.5 T and and E = 1000 V/mE = 1000 V/m. . What is its mass?What is its mass?

mvR

qB qBR

mv

qBR

mv

1000 V/m

0.5 T

Ev

B

v = v = 2000 2000 m/sm/s

-19(1.6 x 10 C)(0.5 T)(0.0728 m)

2000 m/sm m = 2.91 x 10-24

kgm = 2.91 x 10-24

kg

+q

R

Ev

B

+-x x x x x x x x

Photographic plate

m

slitx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x

Summary Summary

N SN

B

vv

FFRight-hand rule

for positive q

N SN

FF

B

vvLeft-hand rule for

negative q

The direction of forces on a charge moving in an The direction of forces on a charge moving in an electric field can be determined by the right-hand electric field can be determined by the right-hand rule for positive charges and by the left-hand rule rule for positive charges and by the left-hand rule for negative charges.for negative charges.

The direction of forces on a charge moving in an The direction of forces on a charge moving in an electric field can be determined by the right-hand electric field can be determined by the right-hand rule for positive charges and by the left-hand rule rule for positive charges and by the left-hand rule for negative charges.for negative charges.

Summary (Continued)Summary (Continued)

B

vv

FF

v sin v sin vv

For a charge moving in For a charge moving in a B-field, the a B-field, the magnitude of the force magnitude of the force is given by:is given by:

F = qvB sin

Summary (Continued)Summary (Continued)

mvR

qBmv

RqB

qBRm

vqBR

mv

x x x x x x

x x

+

-

+qv

VV

Ev

BE

vB

The The velocity velocity selector:selector:

+q

R

Ev

B+-

x x x x x x x x

m

slitx x x x x x x x x x x x x x x x x x

x x x x x x x x

The mass The mass spectrometer:spectrometer: