AP Physics “C” – Chapter 23 Notes – Yockers
Faraday’s Law, Inductance, and Maxwell’s Equations
Faraday’s Law of Induction
- induced current
→a metal wire moved in a uniform magnetic field
- the charges (electrons) in the wire will experience a force
- the resulting movement of charges produces a current in the wire
→a stationary wire loop and moving magnet
- an electric current is set up as long as relative motion occurs between a magnet
and a coil
→current can exist in a wire even when the wire is not connected to an emf
→Faraday’s experiment
- galvanometer reads zero when there is no current or there is a steady current
- the “moment” the switch is closed or opened a current is detected on the
galvanometer
→the current produced in the secondary circuit occurs only for an instant
while the magnetic field acting on the secondary coil builds from its
zero value to its final value (when the switch is closed)
→the changing magnetic field is a result of the current in the primary
circuit
- an electric current can be produced by a time-varying magnetic field
- an emf is induced in the secondary circuit by the changing magnetic field
- magnetic flux  B  – proportional to the number of magnetic field lines passing through an
area
→the magnetic flux through the element is B  dA
- dA is a vector perpendicular to the surface whose magnitude equals the area
dA
- B is the magnetic field at the surface of the area dA
- the SI unit of magnetic flux is the weber: 1 Wb = 1 T·m2
- the total magnetic flux through the surface is
 B   B  dA
→emf is induced in a circuit when the magnetic flux through the surface bounded by the
circuit changes with time
- Faraday’s law of induction – the emf induced in a circuit is equal to the time rate of change of
magnetic flux through the circuit
d B
 
dt
→if the circuit is a coil consisting of N identical and concentric loops and if the field lines
pass through all the loops, then
d B
  N
dt
- the emf is increased by the factor N because all of the loops are in series
- the emfs in the loops add to give the total emf
→if the magnetic field is uniform over the area A bounded by a loop lying in a plane,
then
 B   B  dA   B dA cos   B cos   dA  BA cos 
 B  BA cos 
-  = the angle between B and the direction ┴ to the plane of the loop. Why?
→  B is at a maximum when plane of the loop is ┴ to B
→  B is at a minimum when the plane of the loop is ║ to B
Lenz’s Law (the negative sign in Faraday’s law) – the polarity of the induced emf in a loop is such
that it produces a current whose magnetic field opposes the change in magnetic flux through
the loop. That is, the induced current is in a direction such that the induced magnetic field
attempts to maintain the original flux through the loop
- consistent with the continuity equation for energy
- a bar moving on two parallel rails in the presence of a uniform magnetic field
- a bar magnet moved with respect to a stationary loop of wire
Motional emf – arises from the motion of a conductor through a magnetic field
- conducting bar of length l moves through a uniform magnetic field B
→due to the magnetic force on the electrons, the ends of the wire become oppositely
charged which establishes an electric field in the wire
→the charge at the ends of the conductor builds up until the magnetic force qvB on an
electron in the conductor is balanced by the electric force qE on the electron
F  F
e
- FB  0
qE  qvB
E  vB
V  E  Bv
→a potential difference is maintained across the conductor as long as there is motion
through the field due to charge separation
→if the motion is reversed, the polarity of the potential difference is also reversed
→the induced motional emf in the bar is 
 B  BA  Bx
d B
d
dx
  Bx    B
  Bv
dt
dt
dt
   Bv
→if the resistance of the circuit is R, the magnitude of the induced current is
 Bv
I 
R
R
- conversion of mechanical energy in the moving bar to internal energy in the resistor
→power is equal to the rate at which energy is delivered to the resistor
→power is equal to the power I supplied by the induced emf
→remember, Fapp  FB  Bv
 
B 2  2 v 2  B 2  2 v 2  R  Bv 
2
  
 
 RI R
R
 R R  R 
- the emf generated by a rotating loop
P  Fappv  IB v 
2
→as the loop rotates, the magnetic flux through it changes with time, inducing an emf
and a current
→if the loop rotates with a constant angular speed in a uniform B
-   t and v  r
-   2f
 B  BA cos   BA cos t
- the emf is at a maximum when the plane of the of the loop is parallel to the
magnetic field
d B
d
  N
  NAB cos t   NAB sin t
dt
dt
→the induced emf is varying and is described by a sinusoidal curve (ac voltage)
Induced emfs and Electric Fields
- an electric field is created in a conductor as a result of changing magnetic flux
- an electric field is always generated by a changing magnetic flux, even in free space where
no charges are present
- the induced current in the loop above implies the presence of an induced electric field E that
must be tangent to the loop in order to provide an electric force on the charges around
the loop
→the work done by the electric field on the loop in moving a test charge q once around
the loop is equal to q
→because the magnitude of the electric force on the charge is qE , the work done by
the electric field can also be expressed as qE 2r 
q  qE 2r 
E

2r
→using Faraday’s law and  B  BA  Br 2 for a circular loop
1 d B
1 d
r dB

E

Br 2   
2r dt
2r dt
2 dt
- this can be used to calculate the induced electric field if the time variation of the
magnetic field is known
- this result is also valid in the absence of a conductor or charges
- the emf for any closed path can be expressed as the line integral of E  ds over that
path
d B
   E  ds  
dt
→the induced electric field E in the equation above is a nonconservative field that is
generated by a changing magnetic field
- it is nonconservative because the work it does moving a charge around a
closed path is not zero
- this electric field is very different from an electrostatic field
Self-inductance
- current doesn’t immediately jump from zero to its maximum value  R
→as the current increases with time, the magnetic flux through the loop of the circuit
itself due to the current also increases with time
→the increasing magnetic flux from the circuit induces an emf in the circuit that opposes
the change in the net magnetic flux through the loop of the circuit
→according to Lenz’s law, the induced electric field in the wires must be opposite the
direction of the current
→the opposing emf results in a gradual increase in the current
- self-induction – the changing magnetic flux through the circuit arises from the circuit itself
- the induced emf is, thus, called a self-induced emf
- the self-induced emf is always proportional to the time rate of change of the current
- for a closely spaced coil of N turns of fixed geometry (a toroidal coil or an ideal solenoid), the
proportionality can be expressed as
d B
dI
 L  N
 L
dt
dt
→L is a proportionality constant called the inductance of the coil
- depends on the geometric features of the coil and other physical characteristics
- L can be determined by
N B
L
I
- L can also be written as (usually the defining equation for the inductance of any
coil)
L
L
dI dt
- the SI unit of inductance is the henry (H) → 1 H = 1 V·s/A
- resistance is a measure of opposition to current, whereas inductance is a measure of
opposition to the change in current
RL Circuits
- a circuit containing a coil has a self-inductance that prevents the current from increasing or
decreasing instantaneously
→a circuit element designed to provide inductance in a circuit is an inductor
→it is assumed that the self-inductance for the remainder of the circuit is negligible (for
this course)
→a series RL circuit – as the current increases toward its maximum value, an emf that
opposes the increasing current is induced in the inductor
- as current begins to increase, after the switch is closed, the inductor produces
an emf (sometimes referred to as a “back emf”) that opposes the
increasing current
- the back emf produced by the inductor is
dI
 L  L
dt
→because the current is increasing, dI dt is positive
→  L is negative
→a potential drop occurs from point a to point b in the diagram above
- applying Kirchhoff’s loop rule, beginning at the battery
dI
  IR  L  0
dt
→the potential across the inductor is given a negative sign because its
emf is in the opposite sense to that of the battery
- to obtain a mathematical solution to the Kirchhoff’s loop rule equation

L dI
I 
0
R
R dt
→letting x   R   I , so dx  dI
L dx
x
0
R dt
dx
R
  dt
x
L
→integrating the last equation from an initial instant to some later time t gives
x dx
R t
x
R
x0 x   L 0 dt  ln x0   L t
→where the value of x at t  0 is expressed as x0   R , because I  0 at t  0
ln
x
R
 t
x0
L
x
 e  Rt L
x0
x  x0 e  Rt L

R
I 

R

which is equivalent to
e  Rt L


1  e  Rt L
R
→which is the solution for current from the Kirchhoff’s loop rule equation
I
I t  

1  e 
R
t 
- where the constant  is the time constant of the RL circuit
-   L R and has units of time (seconds)
V  s A V  s  A
L
 

s
R

A V
- the steady-state current value of the current which occurs at t   is
 R
→in steady-state the change in current is zero
→the final current does not involve L because the inductor has no
effect on the circuit if the current is not changing
→the first time derivative of the equation above
dI  t 
 e
dt L
- the rate of current dI dt is a maximum (equal to  L ) at t  0 and falls
exponentially to zero as t  
- an RL circuit containing two switches
→when S1 is closed and S2 is open, the battery is in the circuit
→at the instant S2 is closed, S1 is opened and the battery is no longer part of the circuit
→applying Kirchhoff’s loop rule to the “S2 closed S1 open” situation with the current
initially at its steady-state value  R
dI
IR  L
0
dt
- the solution of this differential equation is
I t  

R
e t   I 0 e  t 
→the current at t  0 is I 0   R and   L R
→the current is continuously decreasing with time
→the slope of dI dt is always negative and has its maximum value at
t 0
→the negative slope signifies that  L   LdI dt  is now positive (point a
in the circuit is at a lower potential than point b)
Energy Stored in a Magnetic Field
- applying Kirchhoff’s loop rule, beginning at the battery
dI
  IR  L  0
dt
- multiplying each term by the current I gives the rate at which energy is supplied by the
battery
dI
I  I 2 R  LI
0
dt
dI
I  I 2 R  LI
dt
2
→ I R is the rate at which energy is delivered to the resistor
dI
→ LI
is the rate at which energy is delivered to the inductor
dt
- if U B denotes the energy stored in the inductor at any time, then the rate dU B dt at which
energy is transferred to the inductor can be written
dU B
dI
 LI
dt
dt
- to find the total energy stored in an inductor at any instant
UB
I
U B   dU B   LI dI
0
0
U B  LI
- an inductor stores energy in its magnetic field when the current is I
1
2
2
Maxwell’s Equations
- can be regarded as the basis of all electric and magnetic phenomena
- represent laws of electricity and magnetism already covered
- predict the presence of electromagnetic waves traveling at c
- show that electromagnetic waves are radiated by accelerating charges
Gauss’s law
Q
 E  dA  
0
→the total electric flux through any closed surface equals the net charge inside that
surface divided by 0
→this law describes how charges create electric fields
→electric field lines originate on positive charges and terminate on negative charges
Gauss’s law for magnetism
 B  dA  0
→the net magnetic flux through a closed surface is zero
- the number of magnetic field lines entering a closed volume must equal the
number leaving
- magnetic field lines cannot begin or end at any point
→isolated magnetic monopoles cannot exist
Faraday’s law of induction
d B
E

d
s



dt
→the line integral of the electric field around any closed path (which equals the emf)
equals the rate of change of magnetic flux through any surface area bounded by
that path
→describes how a changing magnetic field creates an electric field
Ampère’s law (generalized form)
d E
 B  ds   0 I  0  0 dt
→the line integral of the magnetic field around any closed path is determined by the net
current and the rate of change of electric flux through any surface bounded by
that path
→describes how both an electric current and a changing electric field create a magnetic
field
- Lorentz force
F  qE  qv  B
→once the electric and magnetic fields are known at some point is space, the force
those fields exert on some particle of charge q can be calculated
→this force and Maxwell’s equations describe all classical electromagnetic interactions