Review&for&Exam&III€¦ · Review&for&Exam&III “Leftovers” RCcircuits. Right?Hand?Rules F qv...
Transcript of Review&for&Exam&III€¦ · Review&for&Exam&III “Leftovers” RCcircuits. Right?Hand?Rules F qv...
Basic Laws
Basic Definitions
Important Derivations
Examples
+B dl I• =∫ µ0
U B= − •! !µ
dF Idl B! ! != ×
R mvqB
=
Review for Exam III“Leftovers” RC circuits
Right-Hand-RulesBvqF!!!
×= BLIF!!!
×=
Fr!!
×=τ B!!
×= µτ
20 ˆ4 r
rsdIBd ×=
!!
πµ
Basic Laws• Biot-Savart Law:
Direct calculation of B from currents.dB I dl r
r
!! !
=×µ
π0
34
x
Rrθ
θ
P
Idx
RHR
We did the integral but it is messy and takes time … so
Basic Laws• Biot-Savart Law:
direct calculation of B from currents.
• Ampere's Law:calculation of B in cases of high symmetry
dB I dl rr
!! !
=×µ
π0
34
B dl I• =∫ µ0
´
If B “constant” along paththen integral is just the path
B IR
=µπ0
2
Basic Laws• Faraday's Law:
determines the induced E field (or emf) from a changing magnetic flux.
ε = −ddtBΦ
! !E dl d
dtB•∫ = −
Φ
dS
B BΦB B dS≡ •∫! !
Define Flux
Basic question: Does it change in time (ie, increase or decrease?)You will have to figure that out
Lenz's LawThe induced current will appear in such a direction that itopposes the change in flux that produced it.
vB
S Nv
BN S
Change Area of loop Change magnetic fieldthrough loop
Change orientation of loop relative to B
EMF Observed when dΦ/dt is non zeroDirection from Lenz’s Law
Examples
BvqEqF!!!!
×+=
Definitions & Derivations
• Lorentz Force:
Force in an electric field Force in a magnetic field
RHR
E x x x x x x x x xx x x x x x x x xx x x x x x x x xx x x x x x x x x
Examples with Magnetic Field only and a Moving charge:
Magnetic Dipole Moment
Bx
.FFθ
θ
µ
Direction: ⊥ to plane of the loop in the direction the thumb of right hand points if fingers curl in direction of current.
• Torque on loop is then:
• Note: if loop consists of N turns, µ = NAI
Magnitude: µ = AI
τ = AIB sinθ Þ! ! !τ µ= × B
Remember this: The torque always wants to line µ up with B!
Definitions & Derivations
RHR
I
Potential Energy of Dipole
Bx
.FFθ
θ
µ
• Work is required to change the orientation of a magnetic dipole in the presence of a magnetic field.
• Define potential energy U (with zero at position of max torque) corresponding to this work.
∫≡ θτdU Þ BU!!
•−= µ
Remember this: The torque always wants to line µ up with B …which minimizes the potential energy!
Examples with RC circuits
• Charging a capacitor:
• Discharging a capacitor:R
I I
Cε
a
b + +- -
RI I
Cε
a
bε = +Rdq
dtqC
Rdqdt
qC
+ = 0
Loop rule !
Charging DischargingRC
t
2RC
0
0 1 2 3 40
0.5
1
t/RC
Q f( )x
x
-ε/R
I
t
q
0
Cε
0 1 2 3 4
0.5
11
0.0183156
f( )x
40 x
q = C εe -t/RC
I dqdt R
e t RC= = − −ε /
RC
t
q
2RC
00 1 2 3 4
0
0.5
1
t/RC
Q f( )x
x
Cε
I
0 t
ε/R
0 1 2 3 4
0.5
11
0.0183156
f( )x
40 x
( )q C e t RC= − −ε 1 /
I dqdt R
e t RC= = −ε /
Examples
• Orbit of charged particle in uniform magnetic field: x x x x x x
x x x x x xx x x x x x
v
F
B
q
x x x x x xx x x x x xx x x x x x
FvR
R mvqB
=
Examples
• B for straight wire:× R
(Ampere)
• B for ¥ current sheet:
• B for ¥ solenoid: xxx xx•• • ••
ii
xxxx
x
xx
xxxx
i
B IR
=µπ0
2
(Biot-Savart)
x
Rrθ
θ
P
Idx
B = µ0 ni
2
B = µ0 ni
Examples: B from long straight wire
B = µ0 I2 π r
a 2
• Inside the wire: (r < a)
• Outside the wire: (r>a)
B = µ0 I2πr
Examples
• Force on parallel current-carrying conductors of length L:
• Currents in oppositedirections Þ⇒ repulsive force.
• Currents in same direction Þ attractive force.
πd2LIIµBLIF ba0
abb =×=!!! L d
×F
Ib
Ia×
F
• Force on current carrying wire in a magnetic field:
dF Idl B! ! != × RHR
Examples• Current induced by pulling coil through magnetic field:
RLBv
dtd
RRI =
Φ==1εIs the flux changing?
If so, is it increasing? Decreasing?How can you express this quantitatively?
The current through this bar results in a force on the bar
RILBvRvBLFvP 2=⎟
⎠⎞
⎜⎝⎛==
BLIF!!!
×=