a b c d. find currents through resistors loop 1loop 2loop 3 loop 4 i1i1 i2i2 i3i3 i5i5 i4i4 loop 1:...
TRANSCRIPT
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
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
What if v is not perpendicular to B?
BvqFmagnetic
Direction?
Magnitude?v
||v
BqvFmagnetic
Trajectory: helix
Exercise
Fmagnetic qvBsin
𝐹𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐=𝑞 (𝑣⊥+𝑣∥ )×𝐵
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
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