Physics 9HD Final Exam Review
Electromagnetic Induction
1 dΦ B
1 d r r
=−
B ⋅ da
Faraday’s Law: E = −
c dt
c dt ∫
Lenz’s Law: The direction of the induced current is such that its magnetic field acts to oppose
the change in the field which caused the induced current.
Eddy currents are induced in bulk conductors when they are in a changing magnetic field. For
example, such currents will act to oppose the motion of a moving conductor through a magnetic
field.
1 r r r
Motional EMF: E = ∫ (v × B) ⋅ ds
c
Electric generator: Mechanically turn the armature of the DC motor in the magnetic field to
generate a current in the armature windings.
Inductance
Mutual inductance:
M =
Self-inductance: E = − L
dI
dt
N 2Φ B 2 N1Φ B1
dI
dI
=
, since E2 = − M 1 and E1 = − M 2
dt
dt
cI1
cI 2
, L=
NΦ B
cI
sec2
Cgs units for inductance in the equations above are
; to convert to henrys, multiply by
cm
10 −9 c 2
To find self-inductance of particular geometry: (1) find magnetic field by Ampere’s Law, (2)
sec2
find magnetic flux through inductor, (3) multiply by N/cI to find L in
, (4) convert to henrys
cm
by multiplying by 10 −9 c 2 .
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Energy stored in inductor: U = LI 2
2
Energy stored in magnetic field: U = ∫ u dv ,
where energy density per unit volume is u =
Transformer:
1 2
B in vacuum.
8π
V2 N 2
=
; If no losses, amplitudes for rms values are related by V1I1 = V2 I 2
V1 N1
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AC Circuits
Resistor, V=IR
Capacitor, Q=CV
Inductor, V = L
⇒ I =C
dV
dt
dI
dt
1
LC
RLC circuit is a damped harmonic oscillator, where the oscillator loses energy by dissipation of
power in the resistor.
tR
−
1
RL circuit, I= E (1 − e L ) ; time constant τ=L/R
R
Turning differential equations into algebra of complex numbers.
Use i = − 1 to keep track of phase of V or I.
LC Circuit is a simple harmonic oscillator, where resonant frequency is ω =
Circuit Element
Resistor
Inductor
Capacitor
I=VY
Admittance Y (adds for
elements in parallel,
since Y=1/Z
V=IZ
Impedance Z (adds for
elements in series)
1
R
1
−i
=
iωL ωL
iω C
R
iωL
1
−i
=
iωC ωC
If circuit is driven by voltage source V = Vo cos ωt , then I = I o cos(ωt + φ ) ,
V =IZ ,
where Z=X+iY=total impedance of the circuit, X and Y real, and Z = X 2 + Y 2 ,
and the phase difference between I and V is φ, where tan φ =
Y
X
Vo
2
Pav = Vrms I rms cos φ
Vrms =
In RLC series circuit, resonance occurs when ω =
maximum in I.
2
1
because this is minimum in Z , causing
LC
Maxwell’s Equations and Electromagnetic Waves
Ampere’s Law corrected with Maxwell’s Displacement Current:
r r 4π
1 dΦ E 4π r r 1 d r r
I enclosed +
=
Integral Form: ∫ B ⋅ ds =
∫ J ⋅ da + c dt ∫ E ⋅ da
cr
c
dt
c
r
r 4π J 1 ∂E
+
Differential Form: ∇×B =
c c ∂t
Maxwell’s equations with no sources ,
r
Maxwell’s Equations in Differential Form
ρ = 0, J = 0
r
r
∇ ⋅ E = 4πρ
∇⋅E = 0
r
r
∇⋅B = 0
∇⋅B = 0
r
r
r
r 4π J 1 ∂E
r 1 ∂E
∇×B =
+
∇×B =
c c ∂t
c ∂t
r
r
r
1 ∂B
r
∂
B
1
∇×E = −
∇×E = −
c ∂t
c ∂t
Know how to derive the wave equation, starting with Maxwell’s equations in vacuum with no
sources, and applying the following vector identity:
r
r r r
r r r
∇ × (∇ × A) = ∇(∇ ⋅ A) − ∇ 2 A
Speed of light c=3 x 1010 cm/sec
λf = c , ω=ck, since ω=2πf, k=2π/λ
r
r
v v
Plane electromagnetic waves, E=B, E ⊥ B , and the E and B fields are each perpendicular to the
direction of propagation of the wave.
r
r
Polarization is direction of E field. For linearly polarized wave, E is constant, e.g.,
r
r
)
)
E = E o y cos(kx − ωt ), B = E o z cos(kx − ωt )
r
For circularly polarized wave, E vector rotates in circle as a function of both space and time.,
r
)
)
e.g., E = E o [ y cos(kx − ωt ) + z sin( kx − ωt )]
Energy and momentum in electromagnetic waves:
r
c r r
Poynting vector S =
E × B = flux of energy per unit time through cross-sectional area
4π
perpendicular to direction of propagation of the wave=energy/(area-time).
r
S is in the direction of propagation of the wave
cE 2
c < E2 >
1
1 2
Emax Bmax =
Emax = rms =
Intensity of the wave=I=Sav=
8π
8π
4π
4π
Electromagnetic spectrum [from long wavelength (low frequency) to short wavelength (high
frequency)]: Radio & TV waves, Microwaves, Infrared, Visible (Red, Orange, Yellow,
Green, Blue, Indigo, Violet), Ultraviolet, X-rays, Gamma Rays
r r
2
E − B 2 and E ⋅ B are both invariant under a Lorentz transformation.
“A light wave looks like a light wave in any inertial frame of reference.” Purcell p. 342
Of course, we expect this from the postulates of relativity.
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