Quizzes/Exercises: Determine Direction Of emf
Indicate the direction of emf in the following cases: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + |B| increases
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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B
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B
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
|B| decreases
|B| decreases |B| increases
|B| decreases
path outside B
or
Faraday’s Law (Reminder) The emf induced in a “circuit” is proportional to the
time rate of change of magnetic flux through the “circuit”.
Notes: “Circuit”: any closed path
does not have to be real conducting circuit
The path/circuit does not have to be circular, or even planar
A
B
θ
nominal direction of ε
Methods to Change Electric Flux
Change of ΦB emf To change ΦB:
Change B emf produced by an induced E field Change A motional emf Change θ motional emf Combination of above
dtBAd
dtd
uniform
)cos( θε −Φ
−= =B
B
Electric Generators
Exercise: Calculate Inductance of a Solenoid
show that for an ideal solenoid:
(see board)
Reminder: magnetic field inside the solenoid (Ch 28) €
L =µ0N
2Al
Area: A
# of turns: N
Energy in an Inductor Energy stored in an inductor is U= ½ LI2
This energy is stored in the form of magnetic field: energy density: uB = ½ B2/µ0 (recall: uE= ½ ε0E2)
Compare: Inductor: energy stored U= ½ LI2 Capacitor: energy stored U= ½ C(ΔV)2
Resistor: no energy stored, (all energy converted to heat)
Basic Circuit Components Component Symbol Behavior in circuit
Ideal battery, emf ΔV=V+-V- =ε Resistor ΔV= -IR
Realistic Battery (Ideal) wire ΔV=0 (R=0, L=0, C=0) Capacitor ΔV=V- - V+ = - q/C, dq/dt =I Inductor ΔV= - LdI/dt
(Ideal) Switch L=0, C=0, R=0 (on), R=∞ (off) Transformer
Future Topics Diodes, Transistors,…
r ε
Discharging a Capacitor in RC Circuit
discharging €
I(t) = −QRC
e−t /RC
Note the time constant τ=RC
€
q(t) /C + R dq(t)dt
= 0
Energy Stored in a LC Circuit
Reminder Energy stored in: Inductor: energy stored UL= ½ LI2 Capacitor: energy stored UC= ½ C(ΔV)2 = ½ Q2/C
Current And Voltages in a Series RLC Circuit
ΔVmax Sin(ωt +φ)
i= Imax Sin(ωt)
ΔvR=(ΔVR)max Sin(ωt) ΔvL=(ΔVL)max Sin(ωt + π/2) ΔvC=(ΔVC)max Sin(ωt - π/2)
22maxmax )( CL XXRIV −+=Δ
)(tan 1
RXX CL −= −φ
Quiz: Can the voltage amplitudes across each components , (ΔVR)max , (ΔVL)max , (ΔVC)max larger than the overall voltage amplitude ΔVmax ?
€
ΔV = ΔVmax sin(ωt + φ)
Electromagnetic Waves EM wave equations:
Plane wave solutions: E= Emaxsin(kx-ωt+φ) B= Bmaxsin(kx-ωt+φ) Properties:
No medium is necessary. E and B are normal to each other E and B are in phase Direction of wave is normal to both E and B (EM waves are transverse waves) Speed of EM wave: E/B = Emax/Bmax=c Transverse wave: two polarizations possible
same φ, set to be 0
x
y
z
E
B c
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Momentum Carried By EM Waves
EM waves: momentum = energy/c
Radiation Pressure (P):
100% absorption
Δp = p P= S/c
100% reflection
Δp = 2p P= 2S/c €
P = u =Sc
Wavelength and Frequency Because of the wave equation the wavelength of and
frequency of a EM wave in vacuum are related by:
Example: Determine the wavelength of an EM wave of frequency 50 MHz in free space
€
λf = c = 3 ⋅108m /s
€
λ =cf
=3 ⋅108m /s50 MHz
=3 ⋅108m /s5 ⋅107s−1 = 6m
Example: Solar Energy The average intensity of the EM radiation from the
Sun on Earth is S ~ 103 W/m2
What is the average radiation pressure for 100% absorption:
What is the force exerted by EM radiation by the Sun on a surface of 1 m2
€
P =Sc
=103W m2
3 ⋅108ms= 3.3 ⋅10−6 N m2
€
F = PA = 3.3 ⋅10−6 N m2 ⋅1m2 = 3.3 ⋅10−6N