Electromagnetic Induction and Waves
(Chapters 33-34)
• The laws of emf induction: Faraday’s and Lenz’s laws
• Concepts of classical electromagnetism. Maxwell equations
• Inductance
• Mutual inductance M
• Self inductance L. Inductors
• Magnetic field energy
• Simple inductive circuits
• RL circuits
• LC circuits
• LRC circuits
• EM waves
• Maxwell’s equations and EM waves
• Energy and momentum in EM waves: Poynting vector
The other side of the coin…
• We’ve seen that an electric current produces a magnetic field. Should we presume
that the reverse is valid as well? Can a magnetic field produce an electric current?
• Yes, magnetic fields can produce electric fields through electromagnetic induction
• Most of the electric devices that we use as power supplies are electric generators
based on induced emf : these generators convert different forms of energy first into
mechanical energy and then, by induction, into electric energy
• Electromagnetic induction was first demonstrated
experimentally by Michael Faraday
Experimental setup:
• A primary coil connected to a battery
• A secondary coil connected to an ammeter
Observations:
• When the switch is closed, the ammeter reads a current and then returns to zero
• When the switch is opened, the ammeter reads an opposite current and then returns to zero
• When there is a steady current in the primary circuit, the ammeter reads zero
Conclusions:
• An electrical current is produced by a changing magnetic field
• The secondary circuit acts as if a source of emf were connected to it for a short time
• It is customary to say that an induced emf is produced in the secondary circuit by the changing
magnetic field. Let’s look closer at why this happens…
Gauss’s Law – Electric Flux
• In order to quantify the idea of emf induction, we
must introduce the magnetic flux since the emf is
actually induced by a change in magnetic flux rather
than generically by a change in the magnetic field
• Magnetic flux is defined in a manner similar to that
of electrical flux:
Def: The electric flux of a
uniform magnetic field B
crossing a surface A under
an angle θ with respect to
the normal to the surface is
θ
θ
SI
A  Anˆ
B
A
B
If the field varies across the surface, one must
integrate the contribution to the flux for every
element of surface:
B
 B  BA
A
 B  BA cos 
 B   B  dA
Ex: The uniform field lines
penetrate an area A perpendicular
and then parallel to the surface,
produce maximum and zero fluxes
 T  m2  Weber  Wb 
B  0
B
Exercise 1: Gauss’s law can be
reformulated for magnetic sources.
Try to do it yourself: what is the
magnetic flux through a closed
surface around a magnetic source?
A4
Quiz:
1. Given a solenoid, through which of the shown surfaces is the
magnetic flux larger?
A1
A2
A1
A2
A3
A3
A4
2. A long, straight wire carrying a current I is placed along the axis of a cylindrical surface of
radius r and a length L. What is the total magnetic flux through the cylinder?
L
a) 0 IL
I
I
b)
0 I
L
2 r
c) Zero
r
Electromagnetic Induction – Faraday’s Law and Lenz’s Law
• The involvement of magnetic flux in the electromagnetic induction is described
explicitly by
Faraday’s Law: The instantaneous emf induced in a circuit equals
the time rate of change of magnetic flux ΦB through the circuit
dB
 
dt
Comments:
• Since ΦB = BAcosθ the change in the flux, dΦB, can be produced by a change in B,
A or/and θ
• If the circuit contains N loops (such as a coil with N turns)    N d  B dt
• The negative sign in Faraday’s Law is included to indicate the polarity of the
induced emf, which is more easily found using:
Lenz’s Law: The direction of any magnetic induction effect is to oppose the cause
of the respective effect
Ex: Causes of EM induction can be: varying magnetic fields, currents, emfs, or forces
determining a change in flux for instance by varying the area exposed to the magnetic field.
Induced effects may be magnetic fields, emfs, currents, forces, electric fields
Electromagnetic Induction – Applying Lenz’s Law
Magnetic flux through a loop can be varied by moving a bar
magnet in proximity
1. If the bar magnet is moved toward a loop of wire:
• As the magnet moves, the magnetic flux increases with time
• The induced current will produce a magnetic field opposing
the increasing flux, so the current is in the direction shown
• This can be seen as a repelling effect onto the incoming bar by
the loop seen as a magnet
Bind
repels the
incoming North
2. If the bar magnet is moved away from a loop of wire:
• As the magnet moves, the magnetic flux decreases with time
• The induced current will produce a magnetic field helping the
decreasing flux, so the current is in the direction shown
• This can be seen as a attraction effect onto the bar by the loop
seen as a magnet
attracts the
departing North
Bind
Exercise 2: Lenz’s Law
Consider a loop of wire in an external magnetic field Bext. By
varying Bext, the magnetic flux through the loop varies an a
current flows through the wire. Let’s use Lenz’t law to find
the direction of the current induced in the loop when
decreasing
Bext
a) Bext decreases with time, and
increasing
b) Bext increases with time
Bext
Exercises:
3. Emf induced by moving field: A bar magnet is
positioned near a coil of wire as shown in the figure.
What is the direction of the induced magnetic field in the
coil and the induced current through the resistor when
the magnet is moved in each of the following directions.
vb
a) to the right
b) to the left
4. Emf induced by switching field on and off: Find the
direction of the current in the resistor in the figure at
each of the following times.
a) at the instant the switch is closed
b) after the switch has been closed for several minutes
c) at the instant the switch is opened
Problem:
1. Emf induced by reversing field: A wire loop of radius r lies so that an external magnetic
field B1 is perpendicular to the loop. The field reverses its direction, and its magnitude
changes to B2 in a time Δt. Find the magnitude of the average induced emf in the loop during
this time in terms of given quantities.
Motional emf – Across a conductor moving in a magnetic field
a
• We’ve seen that, in certain conditions, motion can determine an
B
+
emf – we are interested in this phenomenon since it stays behind
converting mechanical energy into electric energy
Fm  qvB
• To see how it happens, consider a straight conductor of length
q+ v
L moving with constant velocity v perpendicular on a uniform
L
field B: the electric carriers in the conductor experience a
magnetic force qvB along the conductor, as on the figure
Fe  qE
• Notice that the electrons tend to move to the lower end of the
–
conductor, such that a negative charge accumulate at the base
b
• Consequently, a positive charge forms at the upper end of the conductor, such that,
as a result of this charge separation, an electric field E is produced in the conductor
• Charges build up at the ends of the conductor until the upward magnetic force (on
positive carriers forming a current) is balanced by the downward electric force qE
• The potential difference between the ends of the
Ex: The magnetic field of Earth is
conductor is similar with the potential difference
about 5×10–5 T. Therefore, if a
between the plates of a charged capacitor:
straight 1-m metallic rod is moved
Vab  EL  FL q  vBL
• This motional emf is maintained across the
conductor as long as there is motion
perpendicular on the field with a
speed of 1 m/s, the emf produced
across it ends is about 5×10–5 V
Motional emf – Producing current in a circuit
• Consider now that the moving bar on the previous
slide has a negligible resistance and it slides on rails
connected in a circuit to a resistor R, as in the figure
• As the bar is pulled to the right with a velocity v by
an applied force, the free charges move along the
length of the bar producing a potential difference and
consequently an induced current through R
• The motional emf induced in the circuit acts like a
battery with an emf
+
R

The integral is
around a closed
conductor loop
  v  B   dl
force per unit charge acted on the
element dl of conductor moving in
the external magnetic field B
B
I
v
L
+
–
The charge carriers are
pushed upward by the
magnetic force
  vBL  RI  I  vBL R
• In general, for any conductor moving with velocity
v in a magnetic field B we have an alternative
expression for Faraday’s law:
I
+
I
  vBL
R
I
Motional emf – Explained using Faraday and Lenz’s Laws
• Alternatively, we can look at the same situation but using Faraday and Lenz’s
laws: the changing magnetic flux through the loop and the corresponding induced
emf in the bar result from the change in area of the loop
1. Increasing circuit area:
• The magnetic flux through the loop increases
Bext
• By Lenz’s law the induced magnetic field Bind
v
I
R
ind Fm
must oppose the external magnetic field Bext.
Fapplied
B
ind
• The direction of the current that will create the
induced magnetic field is given by RHR #2.
2. Decreasing circuit area:
• The magnetic flux through the loop decreases
• By Lenz’s law the induced magnetic field Bind
Bext
must “help” the external magnetic field Bext.
Iind
R
dA
v
L
• The direction of the current that is reversed
Bind
compared with the case above.
• Then, by Faraday’s Law, we obtain the same
dx
expression for the induced emf:
d
dA
dx
 
B
dt
B
dt
 BL
dt
 BLv
Problems:
2. Gravity as applied force to induce emf: A metallic rod of mass
m slides vertically downward along two rails separated by a distance
ℓ connected by a resistor R. The system is immersed in a constant
magnetic field B oriented into the page.
a) Calculate the current flowing through the resistor R when the
magnetic force on the rod becomes equal to its weight.
b) Calculate the emf induced across the resistor R.
c) Use Faraday’s Law to compute the speed of the rod when the net
force on it is zero.
m
l
B
R
3. Faraday disk: A thin conducting disk with radius R laying in xy-plane rotates with
constant angular velocity ω around z-axis in a uniform magnetic field B parallel with z. Find
the induced emf between the center and the rim of the disk.
Applications – Electric Generators
• An alternating Current (ac) generator converts mechanical energy to electrical
energy by rotating loops of wire in magnetic fields
• There is a variety of sources that can supply the energy to rotate the loop, including
falling water, heat by burning coal or nuclear reactions, etc.
Basic operation of the generator: as the loop
rotates, the magnetic flux through its surface A
changes with time, such that an emf is induced
• For constant angular speed ω = dθ/dt,
I
r
θ
B
dB
d
d
 
   BA cos    BA sin 
dt
dt
dt
v
   BA sin t   max sin t
v
r
θ = ωt
Comments:
• The emf polarity varies sinusoidally (ac signal)
ω
• ε = εmax when loop is parallel to the field
• ε = 0 when the loop is perpendicular to the field
v
B
Concepts of classical electromagnetism – Induced electric fields
• One of the main concepts that you must retain from our discussion about induction
is that a changing magnetic flux can produce an electric field (manifested as an emf)
even in the absence of a magnetic field in the region where the charges move.
Ex: Recall that varying the current in the inner solenoid will induce a
current in the outer solenoid even though the field outside the inner
solenoid is negligible and the outer solenoid doesn’t move
• An electric field induced by a varying magnetic field flux is called
nonelectrostatic field, and it doesn’t need the presence of a conductor
• Moreover, unlike the electrostatic fields (created by charge
distributions) they are not conservative
increasing B
• For instance, for a stationary loop of induced
current or field, the emf is given by Faraday’s
law:
d
   E  dl  
B
dt
0
E
E
or
E
Eind
E
which means that the work done by the field around the loop is not zero for closed
paths – a property of nonconservative fields…
• Therefore, we cannot associate a potential with this kind of electric field
• However, a charge in such a field is still acted by a force F  qE
Concepts of classical electromagnetism – Displacement current
• In conclusion, electromagnetism stems from the symmetry between electricity and
magnetism: a changing magnetic flux can produce an electric field and a varying
electric field creates a magnetic field.
• This interplay can be seen by looking at Maxwell’s discovery: the displacement
current due to a varying electric flux ΦE
• Consider a parallel plate capacitor: the fact that the
ordinary “conduction” current I (apparently) stops
on the plates can trouble Ampere’s law; for instance,
the two surfaces in the figure have the same
boundary, but the currents are different (I and 0).
• To solve the contradiction, Maxwell assumed the
existence of a continuation of the current inside the
capacitor: the displacement current ID given by the
change in electric flux:
 B  dl    I  I 
0
D encl
dE 

 0  I   0

dt encl

• So, the displacement current has its own
measurable magnetic field
I
I
I
I
Concepts of classical electromagnetism – Maxwell equations
• In 1865, James Clerk Maxwell provided a mathematical theory – know today as
classical electromagnetism – that unified electric and magnetic interactions
• Maxwell’s equations crystallize some of the ideas we’ve discussed in class, showing
that electric and magnetic fields play symmetric, complementary roles in nature:
Thus, for charges and currents in an empty space (no dielectric or magnetic
material present) Maxwell’s equations are:
 E  dA 
qencl
(Gauss’s Law)
0
 B  dA  0
if B varies, the resulting E is not conservative
dB
 E  dl   dt
(Faraday’s Law)
missing terms: no magnetic monopoles
dE 

B

dl


I


0
0


dt

encl
(Ampere’s Law)
electric monopoles: magnetic field is never conservative
Quiz 4. The adjacent magnetic field is decreasing. Which of the fields
below is the induced electric field equivalent to an induced current?
a)
b)
c)
B
d)
Problem:
4. Magnetic field inside a charging capacitor: A
parallel-plate capacitor with circular plates of radius R is
being charged by a constant current I. Find the magnetic
field at an arbitrary distance r from the axis of the
capacitor.
I
I
Inductance – Mutual inductance
• By Faraday’s law, changing a current in a coil induces an emf in an adjacent coil:
this coupling is called mutual inductance
• Consider two coils with N1 and N2 turns.
• The variation of current in the first coil corresponds to a
proportionally varying flux through the second:
M 21 i1  N2 B 2
mutual induction
• By Faraday’s law
dB2
di1
 2   N2
  2   M 21
dt
dt
• The mutual-inductance depends on the geometry of the two coils and on the
presence of a magnetic material as a core. If the material has linear magnetic
properties, the mutual inductance is a constant.
• The discussion is symmetric in the opposite direction, so we have
M 21  M12
N 2 B 2 N1 B1
 M

i1
i2
M  Henry  H   1W A
Inductance – Self inductance
• Notice that nothing prevents a changing flux to produce an emf in the very coil that
produces the actual flux: this phenomenon is called self-inductance: discovered in
the 19th century by Joseph Henry
Ex: Consider a current carrying loop of wire
• If the current increases in a loop, the magnetic flux through the loop surface due to this
current also increases: hence, an emf is induced that opposes the change in magnetic flux
• This opposing emf results in a slowed down increase of the current through the loop
• Alternatively, if the current decreases, the self-inductance will slow down the rate of decrease
• The self-induced emf is proportional to the
rate of change of the current through the coil:
di
  L
dt
negative sign indicates
that a changing current
induces an emf in
opposition to that change
• L is a proportionality constant called the inductance of the coil:
Def: If a circuit with N loops carrying a current I produces a magnetic flux ΦB
through each loop surface, the self-inductance is given by
LN
B
I
L
SI
 Henry (H)
Inductance – Inductors
• L characterizes solenoids as elements of circuit called inductors
• Since the flux is proportional to the current, the inductance of a solenoid does not
depend on the current flowing through the coil: it is a characteristic of the device,
depending on geometric factors and the magnetic properties of the interior of the coil
Ex: Self inductance of a straight solenoid: A straight solenoid
with n turns per length, and volume V has inductance given by:
i
A
0 n I
B
BA
LN
N
N
A  0 n2 A  L  0 n 2V
I
I
I
• Inductance can be interpreted as a measure of opposition to the rate of change in the
current: it determines a potential difference or a back emf across the terminals
εback<0
εback>0
Symbol:
i
• A device with self-inductance (such as a coil) is called an
inductor: a circuit element with a certain inductance – an
additional circuit element besides capacitors and resistors
L
a
b
Potential difference:
di
Vab  Va  Vb  L   back
dt
• drop if di/dt > 0
• raise if di/dt < 0
• 0 if i = const.
Energy Stored in a Magnetic Field – Summary of circuit elements
• The work done by a battery to produce an
increasing current against the back emf of an
inductor can be thought of as energy stored
in the magnetic field inside the inductor
 U L  12 LI 2
I
U L   Pdt   iVab dt  L  idi
0
L  0 n2 A
A
VL   L
• Contrast with the energy dissipated across a current carrying resistor:
PR  12 RI 2
R
A
VR  RI
A
• Or the energy stored in the electric field of a charged capacitor:
U C  12 CV
C  0 A d
A
2
di
dt
VC 
d
Q
C
• The magnetic energy density stored in a straight solenoid inductor is given by
1
2
2
LI
UL
u


V
V
1
2


0 n 2 V I 2
V
1
B2
2

 0 nI  
2 0
2 0
this is, in general,
the magnetic energy
density in vacuum
• Inside a magnetic material – such that an iron core inside a solenoid – μ0 is to be
replaced with μ : magnetic permeability in the respective material
Inductive circuits – LR-circuit: principles
i
ε
S1
• An inductor can be combined in series with a resistor into
a dc-RL circuit to obtain a specific behavior
S2
• Recall that the resistance R is a measure of opposition to
εL<0
the current while the inductance L measures the opposition
S1:
to the rate of change of the current. Let’s see what’s
happening in an RL circuit:
R i
L
1. Close S1 and open S2 : the RL series circuit is completed
S2:
εL>0
across a battery ε
• As the current begins to increase, the inductor produces a negative back emf εL < 0
that opposes the increasing current, so the current doesn’t change from 0 to its
maximum instantaneously
• When the current reaches its maximum, the rate of change and the back emf εL = 0
2. Open S1 and close S2: the RL series circuit is completed with battery removed
• Since there is no battery, the current starts to decay, such that the inductor produces
a positive back emf εL > 0 to “help” the current.
• If the current becomes zero, the rate of change and the back emf are εL = 0
• Kirchhoff rule applies in both cases
(set ε = 0 when current is decaying):
di
  iR   L  0    iR L  0
dt
Inductive circuits – LR-circuit: characteristics
1. The current in the RL circuit in series with a battery
increases exponentially to Imax = ε/R:

i  I max 1  et 
di

  iR  L  0  i  1  e Rt L 
dt
R


• The time constant, τ = L/R, for an RL circuit is the
time required for the current in the circuit to reach
63.2% of its final value
• A circuit with a large time constant will take a longer
time to reach its maximum current
2. The current in the RL circuit without a battery
decays exponentially from its initial value I0:
di
iR  L  0  i  I 0e Rt L 
dt
• If the current reached the maximum value before the
battery was disconnected, it is given by I0 = Imax = ε/R
i  I 0 e t 

Inductive circuits – LC-circuit: principles
• An inductor connected across a charged capacitor form an electric oscillator with
oscillating current and charge called a dc-LC circuit. Functionality:
1. As the capacitor discharges, current increases from 0 to a maximum value and the
potential difference across both elements decreases gradually to 0: the electric energy
is stored in the form of magnetic energy
2. When the current reaches its maximum, the capacitor starts to recharge with an
inverse polarity than initially until the current is again zero and the process restarts in
the reverse direction
electric
energy
magnetic
energy
cycle
Inductive circuits – RL-circuit: characteristics
+q
• Kirchhoff rule can be applied to find the equation describing the
oscillations of charge and current:
Angular frequency ω2 = 1/LC
di q
d 2q
2
vL  vC  0  L   0 


q0
2
dt C
dt
• We see that the charge on the capacitor satisfies
an equation similar with that of a Simple
Harmonic Oscillator with angular frequency
 1
LC
• The SHO solutions are
Charge:
q  Q cos t   
maximum charge
Current:
initial phase angle given
by the charge at t = 0
dq
i
 Q sin t   
dt
q
C
vC 
i
–q
q
C
vL  L
i
di
dt
L
 q  Q cos t  2 
φ = π/2
Q
T/2
3T/2
T
t
2T
–Q
 q  Q cos t 
φ=0
imax
i
t
T
–imax
 di 
 
 dt  max
 di 
 
 dt max
2T
di/dt
t
T
2T
Exercise 5: LC oscillations compared with a mechanical
analog: Recall that in PHY181 we studied the oscillations of a
SHO containing a mass m connected to a light spring of force
constant k oscillating on a frictionless horizontal surface.
k
m
0
x
a) Do you remember the SHO equation in this case?
b) Find the oscillating quantities analogue to electric quantities oscillating in the LC circuit:
LC:
charge q
current i
change in current di/dt
Spring:
Problem:
5. LC oscillator: A power supply with emf ε is used to fully charge up a capacitor C. Then the
capacitor is connected to an inductor L.
a) What is the frequency and period of the LC circuit?
b) Find the maximum charge, the maximum current and the maximum rate of change of
current in the circuit.
c) Write out the time dependency of the charge, current and rate of change of current
considering t = 0 the first time when the capacitor holds only half of its maximum charge.
d) Sketch the q vs t graph.
Electromagnetic Waves – Maxwell’s prediction and Hertz’s confirmation
• Maxwell’s electromagnetism modeled the coupling between electric and magnetic
fields and predicted that varying magnetic and electric fields may continually create
each other and propagate in the form of electromagnetic waves
Ex: radio waves, visible light, X-rays etc.
• Maxwell obtained theoretically the speed of this
propagating EM disturbance (speed of light) created
by an accelerating charge
Ex: A charge oscillating harmonically transmits a
wave: the clip shows the wave transmitted
transversally along a horizontal line of field
• Short time after Maxwell’s prediction, Heinrich Hertz confirmed the
electromagnetic waves experimentally using two LC circuits
• When in resonance, the circuits exchanged energy, and Hertz hypothesized the
energy transfer was in the form of electromagnetic waves
• So, he confirmed Maxwell’s theory by showing the waves existed and had all the
properties of light waves such as experiencing reflection, refraction and diffraction
Electromagnetic Waves – Produced using an Antenna
• When a charged particle is accelerated, it must radiate energy, such that, if currents
change rapidly in a circuit, some energy is transmitted in the form of EM waves
• For instance, an alternating voltage applied to an antenna forces the electric charge
in the antenna to oscillate and transmit an EM wave
Ex: A simple EM-wave emitter
• Consider two rods connected to a source of
t=0
alternating current: charges oscillate
producing a polarization (a) and thus an
electric field E
• As oscillations continue, the rods become
less charged, the E-field near the charges
decreases, while the field produced at t = 0
moves away from the rod (b)
• Then the charge polarization and the
corresponding field reverse (c)
t = T/4
B
t=T
E
E
B
B
B
E
E
a)
t = T/2
b)
c)
d)
• Notice that, due to the change of electric field, a magnetic field B is generated
perpendicular on the electric field E and, as the current changes, the magnetic field spreads
out from the antenna in conjunction with the electric field
• The oscillatory correlated electric and magnetic fields are transmitted in space with speed of
light c, and a frequency given by the frequency of the current oscillation in the antenna
Electromagnetic Waves – Properties
• In conclusion, an electromagnetic wave is formed of
oscillating electric and magnetic fields perpendicular to
each other and perpendicular on the direction of
propagation of the wave: so, the electromagnetic waves
are transverse waves
• The electromagnetic wave produced by a charge
oscillating harmonically produces at a large distance a
sinusoidal wave
ˆ max cos  kx  t 
E  x, t   yE
where
ˆ max cos  kx  t 
B  x, t   zB
• The directions of the E and B vectors are
given by a right hand rule (call it #3):
• EM-waves travel at the speed of light c
which can be written in terms of the electric
and magnetic properties of the medium of
propagation, ε0 and μ0
• Also, the theory predicts that the magnitudes
of the E and B fields are related by:
λ
L
k is the wave number, k = 2π/λ
ω is the angular frequency, ω = 2π/T
c 1
Emax  cBmax
Bmax   0 0cEmax
0 0
Electromagnetic Waves – Energy and momentum. Poynting Vector
• Recalling the energy density carried by the electric and magnetic fields, we obtain
the energy carried by an EM-wave in vacuum:
2
1
1 2 1
1
2
2
u  0E 
B  0E 
 0 0 E   0 E 2
2
2 0
2
2 0
• The elementary amount of energy transported by the
wave front A with speed c per unit area per unit time is
given by the energy contained in a volume Acdt


1 dU 1 udV
dx
EB
2
S

u
  0cE 
A dt
A dt
dt
0
c
Def: The energy flow rate is the magnitude of a vector called Poynting vector
with the direction given by the direction of wave propagation
S
1
0
EB
S
SI

J
W

m 2s m 2
• Therefore, the power imparted by the wave over a certain surface A can be
calculated using
P   S  dA
Electromagnetic Waves – Radiation Pressure
• Since EM-waves carry energy as they travel through space, this energy can be
transferred to objects placed in their path
• Since the component fields are alternating, the energy carried by EM waves is shared
equally by the electric and magnetic fields
• The average power P per unit area A is
2
2
Emax Bmax Ema
c
B
x
called intensity of the wave, given by the
I S 

 max
2 0
2 0 c 2 0
average of the Poynting vector over one cycle:
• Hence, EM-waves transport linear momentum as well as energy. The momentum
delivered to a surface is
dp I
p

1. For complete absorption of energy U,
2. For complete reflection of energy U,
• Therefore, using Newton’s 2nd
law, we see that the force exerted
by EM waves perpendicular onto
surfaces is given by:
Adt c
dp 2 I

Adt c
p
p
IA

complete
absorption:
F


dp
c
F

dt
complete reflection: F  2 IA

c
Problem:
z
6. Radiation pressure: An electromagnetic wave travels in
vacuum vertically downward along z-direction with the
perpendicular EM-fields in the x and y-directions, as in the figure.
The beam strikes perpendicularly a perfectly reflective disk of
radius r and mass m.
r
a) Calculate the intensity of the beam knowing that the disk
accelerates downward with an acceleration a.
b) Show the directions and calculate the maximum amplitudes of
the electric and magnetic fields.
c) What should be the intensity of the beam required to levitate the
disk?
x
y