Review: Self-inductance
Announcements
•WebAssign HW Set 7 due this Friday
• Problems cover material from Chapters 20 and 21
εinduced
• We’re skipping Sections 21.1-21.7 (alternating current
circuits)
Binduced
QUESTIONS? PLEASE ASK!
Review: Self-inductance
„
Self-inductance occurs when the changing flux through a
circuit arises from the circuit itself
„
The self-induced EMF must be proportional to the time
rate of change of the current
ε = −N
„
„
„
„
ΔΦ B
Δt
ε = −L
ΔI
Δt
L is inductance of the device; the negative sign indicates
that a changing current induces an EMF in opposition to that
change
The inductance of a coil depends on geometric factors
The SI unit of self-inductance is the Henry: 1 H = 1 (V · s) /
A
The expression for L is
NΦ B
ΔΦ B
L=N
=
I
ΔI
20.7 RL Circuits
„
Consists of voltage source, a resistor, and an inductor:
„
When the current reaches its maximum, the rate of
change and the back emf are zero
„
The time constant, τ, for an RL circuit is the time
required for the current in the circuit to reach 63.2% of
its final value:
τ =
„
L
R
The current at any time
can be found by
I =
ε
R
(1 − e )
−t / τ
1
Problem: 20.50, p694
20.8 Energy Stored in a Magnetic Field
„
The battery has to do work to produce a
current against the back EMF
„
In the circuit to the right,
ε = 6.00V, L = 8.00 mH,
and R = 4.00 Ω.
„
This work can be thought of as energy stored by the
inductor in its magnetic field –> potential energy!
What is the inductive time
constant of the circuit?
Calculate the current in
the circuit 250 μs after the
switch is closed.
What is the final value of
the steady state current?
How long does it take the
current to reach 80% of
its maximum value?
(a)
„
The energy stored in an inductor is:
PEL = ½ L I
(b)
2
(c)
„
Analogous to energy stored in a capacitor:
E=½
(d)
Q2/C
20.8: Maxwell’s Theory of
Electromagnetism
„
Electricity and magnetism were
originally thought to be unrelated
„
in 1865, James Clerk Maxwell developed
a unified theory of electromagnetism
„
Started from the following observations
(thanks to Gauss and Faraday!)
Chapter 21
Electromagnetic
Waves
(We’re skipping Sections 21.1-21.7)
„
„
„
„
Electric field lines originate on positive
charges and terminate on negative
charges
Magnetic field lines always form closed
loops – they do not begin or end
anywhere
A varying magnetic field induces an EMF
and hence an electric field (Faraday’s
Law)
Magnetic fields are generated by moving
charges or currents (Ampère’s Law)
2
Hertz’s Confirmation:
Generation of Radio Waves
Maxwell Equations
All fundamental information about E&M is
contained in these equations!
Source: Wikipedia
„
Hertz used an LC circuit
„
„
It oscillates! It radiates!
„
Switch closes; current flows to capacitor
„
The capacitor charges fully
„
The capacitor then discharges…
„
… and the current increases
„
„
„
„
„
„
Hertz measured the speed of the
waves from the transmitter
„
„
„
He used the waves to form an
interference pattern and calculated
the wavelength
From v = f λ, v was found
v was very close to 3 x 108 m/s, the
known speed of light
An EMF source is connected to provide current
and then taken out of the circuit
total energy of the circuit is stored in capacitor
(electric field); the current is zero and no
energy is stored in the inductor
energy stored in the electric field decreases
energy stored in the inductor (magnetic field)
increases
When the capacitor is fully discharged, there is
no energy stored in its electric field
When the current is a maximum, all the
energy is stored in the magnetic field in the
inductor
The process repeats in the opposite
direction
„
Hertz Measures the Speed of the
Waves
„
E=½LI2
How does it work?
„
Don’t worry - you are not expected to know this for PHY 2054
(If you want to, you should major in physics!)
E = ½ Q 2/C
The energy ‘sloshes’ around the circuit;
there is a continuous transfer of energy
between the inductor and the capacitor
Why this works - Electromagnetic
Waves Produced by an Antenna
„
When a charged particle accelerates, it
radiates energy
„
„
„
When currents in an AC circuit change rapidly,
energy is radiate in the form of EM waves
EM waves are radiated by any circuit carrying
alternating current
An alternating voltage applied to the wires of
an antenna forces the electric charges in the
antenna to oscillate
3
Properties of EM Waves (stated with
out proof)
Electromagnetic Waves are Transverse
Waves
The E and B fields are
perpendicular to each other
Both fields are perpendicular to
the direction of motion
„
„
„
c =
„
I =
1
Because EM waves travel at a
speed that is precisely the
speed of light, light is an
electromagnetic wave
EM waves are
distinguished by
their frequencies
and wavelengths
„
„
Energy is carried by EM waves; it is shared equally
by the electric and magnetic fields
Average power per unit area =
„
Electromagnetic waves transport momentum as
well as energy
For complete absorption of energy U, p=U/c
For complete reflection of energy U, p=(2U)/c
„
„
Doppler Equation for EM Waves
„
The Doppler effect for EM waves
u⎞
⎛
fo ≈ fs ⎜ 1 ± ⎟
c⎠
⎝
c = ƒλ
Wavelengths for
visible light range
from 400 nm to 700
nm
There is no sharp
division between
one kind of EM
wave and the next
2
Emax Bmax
E2
c Bmax
= max =
2 μo
2 μo c
2 μo
Intensity (I) is average power per unit area
„
μ oε o
The Spectrum of EM Waves
„
„
E&M waves are ‘transverse
waves’
For an E&M wave: c = E
B
Electromagnetic waves travel at
the speed of light
„
„
„
„
„
„
„
fo is the observed frequency
fs is the frequency emitted by the source
u is the relative speed between the source
and the observer
The equation is valid only when u is much
smaller than c
4