Physics 7C, Lecture 5Physics 7C, Lecture 5

Winter Quarter -- 2007Winter Quarter -- 2007

Electric Potential, Magnetism,

Magnetic Forces

Professor Robin Erbacher343 Phy/Geo

erbacher@physics.ucdavis.edu

AnnouncementsAnnouncements

• Course syllabus (policy, philosophy) on the web: http://physics7.ucdavis.edu

• Unit 9 continues today: DLMs 8 -14.

• Quiz #3 today, on optics and electric fields/forces. • Quizzes every other lecture, ~20 minutes each, average of 4 best = 45% (or 20)% of grade.

• Turn off cell phones and pagers during lecture.

Gradient Relations: Potential Energy

Gradient Relations: Potential Energy

Recall: What is the potential energy of a mass m in a the Earth’s gravitational field, a height h above the surface of the Earth?

PE = mgh !

• Force on a mass m in gravity field g is F = mg.• Magnitude of force is the spatial derivative, or gradient, of the potential energy of the mass:

€

Fon m = -d

drPE gravity( )

€

PE gravity = - GMm

r

The direction of the force on the mass m is toward decreasing PEgrav (hence the negative sign!)

Gradientrelation

Gravitational Potential EnergyGravitational Potential Energy

Force increases with greater slope

Pote

ntia

l Energ

y d

iffere

nce

r

-

0

PE = GMm/r

F = - PE/r, the - slope

Negative means rapid decrease of PE with decreasing r

Gradients for E Forces: Potential Energy

Gradients for E Forces: Potential Energy

• Force on a charge q in an Electric field E is F = qE.• Magnitude of force is the spatial derivative, or gradient, of the potential energy of the charge:

€

Fon q = -d

drPE electric( )

€

PE electric = - kQq

r

Fon q due to Q = kQq

r2 = qE

E due to Q = kQ

r2

The direction of the force on the charge +/- q is toward decreasing PEcharge (hence the negative sign again!)

Gradients for E Fields: Electric Potential

Gradients for E Fields: Electric Potential

• Slope of the Electric Potential•Constant with distance•Negative

• Electric field is•Constant as a function of distance•Positive

1 :

ˆ ˆ ˆ3 :

dVD E

dxdV dV dV

D E x y zdx dy dz

→

→

=−

=− − −

Electric Potential V (voltage)

Electric Potential V (volts)Electric Potential V (volts)

• Electric potential V depends on position, and distances.• The electric field E can be determined by the spatial derivative of an electric potential, V.

Equipotential surfaces for point charge: Lines where V is the same.

• Circles are 0.5 volts apart, but distances between are NOT uniform.

• Circles get closer and closer toward center. • Potential grows as 1/r.

• Field lines perpendicular to equipotential surfaces.• Electric Field points in direction of decreasing potential.

VE

s

=−

Electric potential of point charges

Electric potential of point charges

• We lose kinectic energy as we get closer, until we stop and rebound

Positive charge: potential hill Negative charge: potential well

• We gain kinectic energy as we get closer, it pulls us in!

Recall: Electric Field of a DipoleRecall: Electric Field of a Dipole

From Lecture 4:Continuous field lines

Topographical Maps: Lines of Equal Elevation

Topographical Maps: Lines of Equal Elevation

3 Views of Mt. Fuji.

Dipole Equipotentials: A “topo” map for electricityDipole Equipotentials: A

“topo” map for electricity

Which direction does the E field point at P?

a) E=0b) To the rightc) To the leftd) Out of page

PRS question:

P

Surface plot: potential for a dipole

Surface plot: potential for a dipole

+ Chargecauses a “hill”

- Chargecauses a “hole”

Review: force, fields, potential energy, potential

Review: force, fields, potential energy, potential

Force Field Pot. Energy Potential

Force AND field arein the direction ofdecreasing PE and

Newtons N/kg Joules (Nm) Joules/kg(kg m/s2) N/C eV Volts

units

2

2

r

kQE

r

GMg

=

=

r

r rfield

r

1∝

F

rq

r

PEEq

rm

r

PEgm

elecelec

gravgrav

−=

−=

−=

−=

r

r

Magnetic FieldsMagnetic Fields

The sun is not a static object, it sometimes can hurl a large amount of particles out into space. These charged particles would quickly erode our atmosphere, but thankfully, Earth’s magnetic field protects us from this solar “coronal mass ejections”. A large part of what makes Earth hospitable is the magnetic field, it is effectively a shield that protects our atmosphere.

Bar magnetsBar magnets

If allowed, bar magnets will always point north or south.

SN

Bar magnets attract one end of a compass needle and repulse the other.

Magnets have poles that attract and repel.

S N S N

S N SN

Earth’s Magnetic FieldEarth’s Magnetic Field

Magnetic Field LinesMagnetic Field Lines

Magnetic Charge?Magnetic Charge?

Some objects can be picked up by magnets, not all.

Iron, nickel, and cobalt are common magnetic materials, copper, aluminum, glass, and plastic are not.

Ends of an attracted object will be attracted.

What happens when a charged object is brought near a magnet?

a) The south pole goes toward the positive

b) The north pole rotates toward the positive

c) Neither pole is attracted. The magnet won’t rotate

Nature tells us electricity and magnetism are related, but not like this.Magnetism is an entirely new type of phenomena.

There is no such thing as a magnetic charge! (monopole)

PRS question:

Inferring a Field…Inferring a Field…

In 1820, a Dane by the name of Hans Christine Oersted discovered that when you bring a compass near a wire carrying current something very interesting happens.

Compasses near a wire…Compasses near a wire…

Faraday’s “field” Faraday’s “field”

Patterns like these led to Faraday’s concept of the field.

Moving Charges Magnetic Fields!

Moving Charges Magnetic Fields!

B-field vectors

I

If a charge (or charges) move, as in a current I, a magnetic vector field B is always induced around the current.

The Magnetic Field BThe Magnetic Field BWe have now deduced the existence of a magnetic field B in the presence of moving charges (or current) I :

What does B depend on?

What units does it have?

In which direction does it point?

€

B = μ0I

2πr

⎛

⎝ ⎜

⎞

⎠ ⎟Magnitude of

magnetic field:

1/r …and current I

Teslas I

B

Right hand rule

for B field vectors

(RHR1)

r

IB

IB

IdB

enclosed

enclosed

πμ

μ

μ

20

0||

0

=

=

=•

∑∫

r

l

lrr

Ampere’s Law:

B

Direction of B field, RHR1Direction of B field, RHR1

BI

To get direction, use right hand rule #1:

Point thumb of RH along direction of current I. Fingers are now curling in direction of B field.

B field from aboveB field from above

Circles around and weakens with increasing r

Current is going toward you

B-field vectors

B = μI

2πr

B directly depends on strength of current

Finding field directionsFinding field directions

Where does the field vector point at spot P?

1) Into the screen

2) Out of the screen

3) Towards the wire

4) Away from the wire

5) Points down

6) Points up

7) Another direction I

P B

W

PRS question:

Finding field directionsFinding field directions

I

P N

W

PRS question:

Where does the field vector point at spot N?

1) Into the screen

2) Out of the screen

3) Towards the wire

4) Away from the wire

5) Points down

6) Points up

7) Another direction

Electric and Magnetic Field Maps

Electric and Magnetic Field Maps

E field

vectors

B field

vectors

stationary

+ Q line

stationary

+ Q line

moving

+ Q line

E field

vectors

B field

vectors

moving

+ Q line

Recall: Force ModelsRecall: Force ModelsRecall our discussion about contact forces, used when direct action occurs between 2 bodies. We called this: Direct Model of Forces

But when we considered questions of indirect forces:How does Earth exert its gravitational force on the ball while in mid-air?

This was an example of action-at-a-distance, leading toField Model of Forces

€

Object A Object B

exerts force

field Object Bexerts forceObject A field

creates

Magnetic Fields and ForcesMagnetic Fields and ForcesAnalogy to Gravity/Electric fields: Magnetic Fields

We can think about moving charge I (current) exerting a force on a moving charge qv.

€

Fdirect

magnitude = qvμ0I

2πrObject A Object B

exerts force

Direct Model of Forces

€

Ffield = qv ×μ0I

2πr

⎛

⎝ ⎜

⎞

⎠ ⎟ = qv × B

field Object B

exerts forceObject A field

creates

Field Model of Forces

Electric and Magnetic Field Maps

Electric and Magnetic Field Maps

Charge creates a field that puts a force on other charges.

Moving q B field force on other moving charges, qv

Moving charge, or a current of moving charge, creates a field that places a force on moving charge. This is a different field than the static E-Field and creates a different type of force. This is an empirical finding.

All charges create E-fields but only moving charges create B fields!

Force due to B FieldForce due to B Field

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

The force on a moving charge dueto a B field is:

€

Ffield = qv ×μ0I

2πr

⎛

⎝ ⎜

⎞

⎠ ⎟ = qv × B

What direction is the resulting force? (What is this cross-product thing?)

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

The Hall Effect (1897)

v

F

B

F

v

B

F

v

B

θ

F = 0

v

Bθ

€

A ˆ x × B ˆ y = C ˆ z

€

qv ˆ x × B ˆ y = Fˆ z

€

Magnitude Ffield = qvBsinθ

Right-hand rule 2!

Finding Magnetic ForceFinding Magnetic Force

I

P N

W

PRS question:

What direction is the force F on a charge +q with velocity v at point N?

1) Into the screen

2) Out of the screen

3) Towards the wire

4) Away from the wire

5) Points down

6) Points up

7) Another direction

+qv

Electrons as ParticlesElectrons as ParticlesWe know that moving electric charges cause magnetic fields. Another source of magnetism can be “spin”.

• Electrons orbiting nuclei create current loops• Protons and electrons themselves have rotation: Spin!

The electron is a source of both

an electric field (due to its

negative charge) and a

magnetic field (due to its "spin")

E B

e ﾐ e ﾐe- e-

1s

2s

2p

3s

3p

4s

3d

Hund's Rules for Fe 2+

2

2

6

2

6

2

4

Hund’s Rules for electron shells levels:Electrons pair up and cancel out magnetic Properties. Leftover electrons can giveMore magnetic properties, like with Fe2+.

Harmonic Waves and LightHarmonic Waves and LightElectric and magnetic fields are everywhere surrounding charges. If we send them into simple harmonic motion, the fields fluctuate in a spatially and time-varying way.

This is light! An electromagnetic (EM) wave!

E ( z ) at a particular t

x �

y �

z �

y

€

Ey z ,t( ) = Ey, max sin2πt

Τ±

2πz

λ+ψ

⎛

⎝ ⎜

⎞

⎠ ⎟,

Bx z ,t( ) = Bx, max sin2πt

Τ±

2πz

λ+ψ

⎛

⎝ ⎜

⎞

⎠ ⎟.

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Alternating Currents (AC)Alternating Currents (AC)As you see in DLM 11, you induce a current by movinga loop through a B field. The current acts to oppose a change in B field through the loop.

Lens’ Law! The current changes direction as you move in and out.

B field

wire loop

(a)

B field

(b)

B field

(c)

Our regular household power is 110V AC. (Can get 220V, multi-phasic, etc). What does DC mean?