A B

C D

Find currents through resistors

Loop 1

Loop 2

Loop 3

Loop 4

I1

I2

I3

I5I4

loop 1:017441111 IRIRIRIremf

loop 2:

033262222 IRIRemfIrIR

loop 3:

0553344 IRIRIR

nodes:04321 IIII

0523 III

0154 III

Five independent equations and five unknowns

Exercise: A Complicated Resistive Circuit

Chapter 21

Magnetic Force

20 ˆ

4 r

rvqB

20 ˆ

4 r

rlIB

The Biot-Savart law for a moving charge

The Biot-Savart law for a short piece of wire:

How does magnetic field affects other charges?

Magnetic Field of a Moving Charge

Direction of the magnetic force depends on: the direction of B the direction of v of the moving charge the sign of the moving charge

BvqFmagnetic

q – charge of the particlev – speed of the particleB – magnetic field

mA

N

m/sC

NT

Magnetic Force on a Moving Charge

BvqFmagnetic

Right Hand Rule for Magnetic Force

Electron charge = -e: The magnetic force on a moving electron is in opposite direction to the direction of the cross product

BvqFmagnetic

What is the effect on the magnitude of speed?

0 ldF

Kinetic energy does not change

Magnetic field cannot change a particle’s energy!

Magnetic field cannot change a particle’s speed!

Magnetic force can only change the direction of velocity but not its magnitude

Effect of B on the Speed of the Charge

BvqFmagnetic

Single electron in uniform B:

sinqvBdt

pd

sinvB

m

q

dt

vd

(v<<c)

e/me = 1.78.1011 C/kg

m/s ; = (1.78.1011 C/kg) m/s)() = 5.3 m/s2

Magnitude of the Magnetic Force

Same as acceleration due to an E= V𝑓𝑜𝑟 𝑣=0?

BvqFmagnetic

Confined area: deflection

What if we have large (infinite) area with constant Bv

qvBdt

pd

Motion in a Magnetic Field

Any rotating vector:

Xdt

Xd

BvqFmagnetic

pdt

pd

ovBqBvqdt

pd90sin

vBqcv

mv

22 /1

22 /1 cvm

Bq

Circular Motion at any Speed

…angular speed

Cyclotron Frequency

22 /1 cvm

Bq

if v<<c:m

Bq

Alternative derivation:

maF Circular motion:

R

va

2

R

v

q vBsin90o m

v2

RmBq

m

Bq

Period T:T

2

Bq

mT 2

Circular Motion at Low Speed

independent of v!

Non-Relativistic

Position vector r: rvdt

rd

r

v

pr

vp

dt

pd

vBq

dt

pd

vBqpr

v

Brqp valid even for relativistic speeds

Used to measure momentum in high-energy particle experiments

Determining the Momentum of a Particle

Circular motion

Brqp

Vqmv

2

2

m

qVv 22

22

2

rB

V

m

e

Determining e/m of an Electron

𝑣=𝐵𝑟|𝑞|/𝑚

|𝑞|𝑚

= 𝑣2

2∆𝑉

|𝑞|𝑚

=(𝐵𝑟|𝑞|/𝑚)2

2∆𝑉

|𝑞|𝑚

=2∆𝑉(𝐵𝑟 )2

1897: m/e >1000 times smaller than H atom

Joseph John Thomson (1856-1940)

Clicker Question

Clicker Question

What if v is not perpendicular to B?

BvqFmagnetic

Direction?

Magnitude?v

||v

BqvFmagnetic

Trajectory: helix

Exercise

Fmagnetic qvBsin

𝐹𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐=𝑞 (𝑣⊥+𝑣∥ )×𝐵

Which direction is electron going?

BvqFmagnetic

Exercise: Circular Motion

Can combine electric and magnetic forces:

BvqFm

EqFe

BvqEqF

Coulomb law and Biot-Savart law have coefficients 1/(40) and 0/(4) to make the field and force equations consistent with each other

The Lorentz Force

BvqEqF

Is it possible to arrange E and B fields so that the total force on a moving charge is zero?

BFB

FE

EBvE

E vB

What if v changes?

B

Ev

A Velocity Selector

𝐹=𝑞 (𝐸+𝑣×𝐵)

A negative charge is placed at rest in a magnetic field as shown below. What is the direction of the magnetic force on the charge?

A. Up

B. Down

C. Into the page

D. Out of the page

E. No force at all.

B

A. Up

B. Down

C. Into the page

D. Out of the page

E. No force at all.

A negatively charged particle is moving horizontally to the right in a uniform magnetic field that is pointing in the same direction as the velocity. What is the direction of the magnetic force on the charge?

B𝒗

A. Left

B. Up

C. Down

D. Into the page

E. Out of the page

Now, another negatively charged particle is moving upward and to the right in a uniform magnetic field that points in the horizontal direction. What is the direction of the magnetic force on the charge?

B

𝒗

Current: many charges are movingSuperposition: add up forces on individual charges

Number of moving charges in short wire:

lnA

Total force:

BvqFm

I

Force of a short wire: BlIFm

In metals: charges q are negative. Will this equation still work?

Magnetic Force on a Current-carrying Wire

𝐹𝑚=(𝑛𝐴∆ 𝑙 )𝑞 𝑣×𝐵¿ (𝑛𝑞𝐴𝑣 )∆ 𝑙×𝐵

v

B

E

????

+-

V>0

Hall Effect

h

When does it reach equilibrium?

|𝐸𝑒⊥|=𝑣|𝐵|

∆𝑉=|𝐸𝑒⊥|h=𝑣|𝐵|h

v

B

E

>0

+-

V>0

v

B

E

????

+-

V<0

Hall Effect for Opposite Charges

By measuring the Hall effect for a particular material,we can determine the sign of the moving particles

that make up the current

Hall Effect

Edwin Herbert Hall(1855 - 1938)

What is the magnitude of the Hall effect in a metal?

Measure know the charge (e)Then we can find n

Monovalent metals: n is the same as # of atoms per m3

Some metals: n is larger than# of atoms per m3

Hall Effect in a Metal

𝐼=|𝑞|𝑛𝐴𝑣𝑣=𝐼

|𝑞|𝑛𝐴  

∆𝑉=h𝐼𝐵

|𝑞|𝑛𝐴  V

Voltmeter 1 reading is POSITIVEVoltmeter 2 reading is POSITIVE

Mobile charges are:

A) Positive (holes)B) Negative (electrons)

C) Not enough information

Clicker Question