Wednesday, July 5, 2006

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Chapter 23:  Magnetic Flux and Faraday's Law of Induction

23-1: Induced Electromotive Force
    A changing magnetic field can produce a current in a circuit due to an induced electromotive force.
    The magnitude of the induced emf (and resulting current) depends on the rate of change of the magnetic flux.
Demo:  Induced emf (coil, magnet, large galvanometer) - Observe that the amount of deflection of the galvonometer needle (which also operates on magnetic induction) depends on how quickly the magnet moves relative to the coil.  The direction of the induced current changes when any one of the following conditions are reversed:  orientation of the magnet, orientation of the coil, relative motion of magnet and coil.  The direction can be predicted by Lenz's Law (see below).

23-2:  Magnetic Flux
    Magnetic flux (like electric flux) is defined as the number of magnetic field lines that cross a given area.
    Phi = BAcos(theta), where theta is the angle between B and the normal to the area.
    Measured in webers:  1 Wb = 1 T-m^2
    Example:  The magnetic flux through a 1 m^2 table top where Bearth = 0.6 G at 70 degrees from horizontal is (0.6e-4 T)(1 m^2)cos(20) = 0.06 mWb

23-3:  Faraday's Law of Induction
    The emf induced by a changing magnetic flux through a coil of N turns is:  emf = -N(dPhi/dt)
    Prob.10 - For a magnetic flux that changes sinusoidally, when is the induced emf a maximum and minimum?
    Prob.16 - Find the average induced emf for a loop of wire 1.12 m long that changes shape from a square to a circle in 4.25 s where B = 0.105 T perpendicular to the plane of the loop.

23-4:  Lenz's Law
    An induced current flows in the direction that opposes the change that caused the current.  (Think of this natural attempt to conserve magnetic flux as the action-reaction law for electromagnetism.)
    Demo: Eddy current brake
    Demo:  Eddy current tube and ramp
    Prob.24 - What is the direction of the current induced in the circuit shown (Fig.23-31) when the magnetic field increases?  Does the top or bottom plate of the capacitor become positively charged?
    Prob.26 - What emf is induced in a coil that encircles a long current-carrying wire if the current in the wire is (a) constant, or (b) increases?  What if the wire is still perpendicular to the plane of the coil but does not pass through its center?
    Demo:  Jumping ring

23-5:  Mechanical Work and Electrical Energy
    Mechanical work can be converted to electrical energy by means of a magnetic field.
    A conductor of length L that is forced to move with speed v through a magnetic field B will have an induced emf = BvL.
    Prob.29,62 - The emf induced in long conductors moving through weak magnetic fields can be significant. (Remember the failed NASA experiment to generate electricity in space using a long wire?)
    CQ: 16 - Describe motion of rod sliding on rails in magnetic field when the switch is closed.

23-6:  Generators and Motors
    An electric generator uses mechanical work to produce electrical energy:  emf = NBAwsin(wt)
    A motor is basically an electric generator operated in reverse.
    Demo - simple motor
    Prob.36 - What is the maximum emf for a coil made from 1.6 m of wire wound into a coil that is 3.2 cm in diameter and rotated at 95 rpm in a 0.070 T magnetic field? [18 mV]
    Prob.37 (assigned for HW) - Which component of the earth's magnetic field contributes to the induced emf in a circular coil rotating about a vertical axis?

23-7:  Inductance
    Self-inductance (or simply inductance, L) is defined in terms of the emf that opposes the change in current within a coil:  emf = -L(dI/dt)
    The SI unit of inductance is the henry:  1 H = 1 V-s/A
    The inductance of a solenoid is:  L = uo*n^2*A*length, where n = N/length
    Prob.44 - Find the inductance of a solenoid that has an induced emf of 75 mV when dI/dt = 2.0 A/s.[38 mH]  What will be the induced emf if the solenoid is stretched to twice its original length? [38 mV]

23-8:  RL Circuits
    The characteristic time over which current changes in an RL circuit is:  tau = L/R
    When the switch is closed in an RL circuit, there is a large self-induced back-emf due to the sudden change in current, so the current through the inductor is initially small, but increases logarithmically to a maximum steady-state value:  I = V/R(1 - e^-tR/L).
    Prob.46 - Find the time constant for the circuit shown in Fig. 23-34. [0.4 ms]  What is the current when t = 2*tau? [60 mA]   What is current after a long time? [70 mA]

23-9:  Energy Stored in a Magnetic Field
    Energy can be stored in a magnetic field (just like energy stored in an electric field).
    An inductor with current I has energy:  U = (1/2)LI^2
    The energy density of a magnetic field is:  u(B) = B^2/(2uo)
    Prob.54 - Find the energy stored in the inductor at t = 0, tau, and infinity.
    CQ20 - What happens to the energy of a solenoid if n is doubled and I is halved?
    P.80 - What is the value of E/B for electric and magnetic fields with the same energy density? [c]

23-10:  Transformers
    A transformer uses magnetic induction to convert current and voltage in one circuit to a different current and voltage in another circuit with minimal power loss.
     The emf in either the primary or secondary coil is proportional to the number of turns in the coil based on Faraday's law of induction. Vp/Vs = Np/Ns.
    All transformers are fundamentally similar, but differ in size according to the maximum power that they can handle without overheating.
    Demo:  Transformer with light bulb
    Ponderable:  Why is the voltage stepped up from 12 kV to 240 kV for transmission of electrical energy over long distances?

Applications of magnetic induction:  generators, motors, transformers, audio speakers, telephone receivers and speakers, read/write heads for magnetic media (tape recorders, VCR, disk drives), tuning circuits for wireless EM devices (radios, cell phones, TVs), electric guitar pickups, MRI, regenerative breaking systems, metal detectors, magnetic anti-theft devices, traffic signal loops in pavement, rechargeable toothbrushes.

Concept Tests

Chapter 24:  AC Circuits

24-1:  Alternating Voltages and Currents
    An ac generator produces a voltage that varies over time:  V = Vmax*sin(wt)
    The voltage read by an AC voltmeter is Vrms = Vmax/sqrt(2) = 0.71*Vmax
    Example:  What is the peak voltage (Vp) for 120 VAC? [170 V]   What is the peak-to-peak voltage (Vpp) for 120 VAC? [340 V]
    Prob.7 - What is Vrms for a square wave?

24-2:  Capacitors in AC Circuits
    The current through a capacitor depends on the frequency and is out of phase with the applied voltage.
    When the voltage across the capacitor is a maximum value, the current is zero, so the voltage lags the current by 90 deg.
    In a phasor diagram, the voltage of a capacitor is drawn at an angle that is 90 degrees clockwise from the current phasor.
    The current through the capacitor can be found from I = V/Xc, where Xc = 1/wC is the capacitive reactance.
    Prob.14 - What is Irms for a pure capacitive AC circuit?

24-3:  RC Circuits
    Impedance is the overall opposition to the flow of current in a circuit due to resistance, capacitance, and inductance.
    Z = sqrt(R^2 + (XL - Xc)^2)

24-4:  Inductors in AC Circuits
    The voltage across an inductor results from the induced emf due to a changing current through the inductor.
    The voltage across the inductor is maximum when the current is a minimum (but maximum slope), so the voltage leads the current by 90 deg.
    In a phasor diagram, the voltage of across an inductor is drawn at an angle that is 90 degrees counterclockwise from the current phasor.
       Tip:  You can remember that V(L) points to the left since it is on the left side of I.
    The current through the inductor can be found from I = V/XL, where XL = wL is the inductive reactance.
    Prob.44 - What is Irms at high and low frequencies?

24-5:  RLC Circuits
    Phase angle - remember:  ELI the ICE man
    Prob.64 - Is fo more or less than 60 Hz based on the phasor diagram?  What is the impedance at resonance?

24-6:  Resonance in Electrical Circuits
    RLC circuits have natural frequencies, similar to a pendulum or a mass on a spring.
    w = 1/sqrt(LC)
    Prob.59 - If L and C are both doubled, what happens to the resonance frequency?
    Demo:  RLC circuit with digital oscilloscope

Concept Tests - Ch. 21