General electric flux definition

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General electric flux definition

• The field is not uniform
• The surface is not perpendicular to the field

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Magnetic Flux
Whenever possible
Units Weber WbT-m2
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Gausss Law
Gauss asserts that the calculation for the flux
through a closed surface from a point charge is
true for any charge distribution!!!
This is true so long as Q is the charge enclosed
by the surface of integration.
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Gausss Law for magnetism
This is true because we cannot isolate a magnetic
pole and because magnetic field lines are
continuous. The net number of field lines
passing through any surface is always zero!
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Faradays Law
When the magnet moves, a current is induced as if
there was a source of emf (like a battery) in the
circuit!
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Active Figure 31.1
(SLIDESHOW MODE ONLY)
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Faradays Law
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Active Figure 31.2
(SLIDESHOW MODE ONLY)
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Faradays Law of Induction

• An induced emf is produced by a changing magnetic
  field.
• Lenzs Law An induced emf is always in a
  direction that opposes the original change in the
  flux that caused the emf.

Units Volts
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How can we change the flux?

• Change flux by
• Change area
• Change angle
• Change field

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30. In Figure P31.30, the bar magnet is moved
toward the loop. Is Va Vb positive, negative,
or zero? Explain.
Figure P31.30
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Faradays Law continued

• Remember FB is the magnetic flux through the
  circuit and is found by
• If the circuit consists of N loops, all of the
  same area, and if FB is the flux through one
  loop, an emf is induced in every loop and
  Faradays law becomes

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Decaying uniform magnetic field P31.4
Assume we can change field
A
R

• Change flux by
• Change field
• Change area
• Change angle

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Applications of Faradays Law Pickup Coil

• The pickup coil of an electric guitar uses
  Faradays law
• The coil is placed near the vibrating string and
  causes a portion of the string to become
  magnetized
• When the string vibrates at the same frequency,
  the magnetized segment produces a changing flux
  through the coil
• The induced emf is fed to an amplifier

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1. A 50-turn rectangular coil of dimensions 5.00
cm 10.0 cm is allowed to fall from a position
where B 0 to a new position where B 0.500 T
and is the magnetic field directed perpendicular
to the plane of the coil. Calculate the magnitude
of the average emf that is induced in the coil if
the displacement occurs in 0.250 s.
3. A 25-turn circular coil of wire has diameter
1.00 m. It is placed with its axis along the
direction of the Earths magnetic field of 50.0
µT, and then in 0.200 s it is flipped 180. An
average emf of what magnitude is generated in the
coil?
6. A magnetic field of 0.200 T exists within a
solenoid of 500 turns and a diameter of 10.0 cm.
How rapidly (that is, within what period of time)
must the field be reduced to zero, if the average
induced emf within the coil during this time
interval is to be 10.0 kV?
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Linear Generator
Charges stop moving when
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Linear Generator with Faradays Law
By Lenzs Law, what is the direction of current?
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Active Figure 31.10
(SLIDESHOW MODE ONLY)
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Power moving the bar
Same result!
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Breaking effect if power not added
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12. A 30-turn circular coil of radius 4.00 cm
and resistance 1.00 O is placed in a magnetic
field directed perpendicular to the plane of the
coil. The magnitude of the magnetic field varies
in time according to the expression B 0.010 0t
0.040 0t2, where t is in seconds and B is in
tesla. Calculate the induced emf in the coil at t
5.00 s.
19. An automobile has a vertical radio antenna
1.20 m long. The automobile travels at 65.0 km/h
on a horizontal road where the Earths magnetic
field is 50.0 µT directed toward the north and
downward at an angle of 65.0 below the
horizontal. (a) Specify the direction that the
automobile should move in order to generate the
maximum motional emf in the antenna, with the top
of the antenna positive relative to the bottom.
(b) Calculate the magnitude of this induced emf.
22. A conducting rod of length l moves on two
horizontal, frictionless rails, as shown in
Figure P31.20. If a constant force of 1.00 N
moves the bar at 2.00 m/s through a magnetic
field B that is directed into the page, (a) what
is the current through the 8.00-O resistor R? (b)
What is the rate at which energy is delivered to
the resistor? (c) What is the mechanical power
delivered by the force Fapp?
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Rotating Generators and Faradays Law
0
For N loops of wire
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Induced emf in a Rotating Loop

• The induced emf in the loop is
• This is sinusoidal, with emax NABw

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Active Figure 31.21
(SLIDESHOW MODE ONLY)
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Rotating Generators
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DC Generators

• The DC (direct current) generator has essentially
  the same components as the AC generator
• The main difference is that the contacts to the
  rotating loop are made using a split ring called
  a commutator

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Active Figure 31.23
(SLIDESHOW MODE ONLY)
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32. For the situation shown in Figure P31.32,
the magnetic field changes with time according to
the expression B (2.00t3 4.00t2 0.800)T,
and r2 2R 5.00 cm. (a) Calculate the
magnitude and direction of the force exerted on
an electron located at point P2 when t 2.00 s.
(b) At what time is this force equal to zero?
45. A proton moves through a uniform electric
field E 50.0 j V/m and a uniform magnetic field
B (0.200i 0.300j 0.400k)T. Determine the
acceleration of the proton when it has a velocity
v 200 i m/s.
59. A circular loop of wire of radius r is in a
uniform magnetic field, with the plane of the
loop perpendicular to the direction of the field
(Fig. P31.59). The magnetic field varies with
time according to B(t) a bt, where a and b
are constants. (a) Calculate the magnetic flux
through the loop at t 0. (b) Calculate the emf
induced in the loop. (c) If the resistance of the
loop is R, what is the induced current? (d) At
what rate is energy being delivered to the
resistance of the loop?
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Eddy Currents
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Self-Inductance

• When the switch is closed, the current does not
  immediately reach its maximum value
• Faradays law can be used to describe the effect

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Self-induced emf

• A current in the coil produces a magnetic field
  directed toward the left (a)
• If the current increases, the increasing flux
  creates an induced emf of the polarity shown (b)
• The polarity of the induced emf reverses if the
  current decreases (c)

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Self Inductance
Define Self Inductance
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Inductance of a Solenoid

• The magnetic flux through each turn is
• Therefore, the inductance is
• This shows that L depends on the geometry of the
  object

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Inductance Units
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3. A 2.00-H inductor carries a steady current of
0.500 A. When the switch in the circuit is
opened, the current is effectively zero after
10.0 ms. What is the average induced emf in the
inductor during this time?
5. A 10.0-mH inductor carries a current I Imax
sin ?t, with Imax 5.00 A and ?/2p 60.0 Hz.
What is the back emf as a function of time?
7. An inductor in the form of a solenoid
contains 420 turns, is 16.0 cm in length, and has
a cross-sectional area of 3.00 cm2. What uniform
rate of decrease of current through the inductor
induces an emf of 175 µV?
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LR Circuits
Charging
Kirchhoff Loop Equation
Solution
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LR Circuits
Discharging
Kirchhoff Loop Equation
Solution
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Active Figure 32.3
(SLIDESHOW MODE ONLY)
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14. Calculate the resistance in an RL circuit in
which L 2.50 H and the current increases to
90.0 of its final value in 3.00 s.
20. A 12.0-V battery is connected in series with
a resistor and an inductor. The circuit has a
time constant of 500 µs, and the maximum current
is 200 mA. What is the value of the inductance?
24. A series RL circuit with L 3.00 H and a
series RC circuit with C 3.00 µF have equal
time constants. If the two circuits contain the
same resistance R, (a) what is the value of R and
(b) what is the time constant?
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Energy in a coil
PE in an Inductor
PE in an Capacitor
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Energy Density in a coil
PE in an Inductor
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31. An air-core solenoid with 68 turns is 8.00
cm long and has a diameter of 1.20 cm. How much
energy is stored in its magnetic field when it
carries a current of 0.770 A?
33. On a clear day at a certain location, a
100-V/m vertical electric field exists near the
Earths surface. At the same place, the Earths
magnetic field has a magnitude of 0.500 104 T.
Compute the energy densities of the two fields.
36. A 10.0-V battery, a 5.00-O resistor, and a
10.0-H inductor are connected in series. After
the current in the circuit has reached its
maximum value, calculate (a) the power being
supplied by the battery, (b) the power being
delivered to the resistor, (c) the power being
delivered to the inductor, and (d) the energy
stored in the magnetic field of the inductor.
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Example 32-5 The Coaxial Cable

• Calculate L for the cable
• The total flux is
• Therefore, L is
• The total energy is

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Mutual Inductance
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Mutual Inductance example
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Induced emf and Electric Fields

• An electric field is created in the conductor as
  a result of the changing magnetic flux
• Even in the absence of a conducting loop, a
  changing magnetic field will generate an electric
  field in empty space
• This induced electric field is nonconservative
• Unlike the electric field produced by stationary
  charges
• The emf for any closed path can be expressed as
  the line integral of E.ds over the path

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General form of Faradays Law
So the electromotive force around a closed path
is
And Faradays Law becomes
A changing magnetic flux produces an electric
field.
This electric field is necessarily
non-conservative.
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E produced by changing B
How about outside ro ?
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Maxwells Equations

• The two Gausss laws are symmetrical, apart from
  the absence of the term for magnetic monopoles in
  Gausss law for magnetism
• Faradays law and the Ampere-Maxwell law are
  symmetrical in that the line integrals of E and B
  around a closed path are related to the rate of
  change of the respective fluxes

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• Gausss law (electrical)
• The total electric flux through any closed
  surface equals the net charge inside that surface
  divided by eo
• This relates an electric field to the charge
  distribution that creates it
• Gausss law (magnetism)
• The total magnetic flux through any closed
  surface is zero
• This says the number of field lines that enter a
  closed volume must equal the number that leave
  that volume
• This implies the magnetic field lines cannot
  begin or end at any point
• Isolated magnetic monopoles have not been
  observed in nature

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• Faradays law of Induction
• This describes the creation of an electric field
  by a changing magnetic flux
• The law states that the emf, which is the line
  integral of the electric field around any closed
  path, equals the rate of change of the magnetic
  flux through any surface bounded by that path
• One consequence is the current induced in a
  conducting loop placed in a time-varying B
• The Ampere-Maxwell law is a generalization of
  Amperes law
• It describes the creation of a magnetic field by
  an electric field and electric currents
• The line integral of the magnetic field around
  any closed path is the given sum

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The Lorentz Force Law

• Once the electric and magnetic fields are known
  at some point in space, the force acting on a
  particle of charge q can be calculated
• F qE qv x B
• This relationship is called the Lorentz force law
• Maxwells equations, together with this force
  law, completely describe all classical
  electromagnetic interactions

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Voltage transformers
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Current transformers
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Example transformers
What is Vs ?