Alternating Current Physics 102 Professor Lee Carkner Lecture 23

PAL #22 RL Circuits  Solenoid: 5 cm long, 1 cm diameter  0.1 V of emf is induced by increasing the current from 0 to 3 A in 0.5 seconds   = -L(  I/  t)  L =  t/  I = [(0.1)(0.5)]/(3)=  L =  0 N 2 A/l  N = (Ll/  0 A) ½  N = [(0.0167)(0.05) / (4  X10 -7 )(  )(0.005) 2 ] ½  N =

A switch is closed, starting a clockwise current in a circuit. What direction is the magnetic field through the middle of the loop? What direction is the current induced by this magnetic field? A)Up, clockwise B)Down, clockwise C)Up, counterclockwise D)Down, counterclockwise E)No magnetic field is produced

The switch is now opened, stopping the clockwise current flow. Is there a self- induced current in the loop now? A)No, since the magnetic field goes to zero B)No, self induction only works with constant currents C)Yes, the decreasing B field produces a clockwise current D)Yes, the decreasing B field produces a counterclockwise current E)Yes, it runs first clockwise then counterclockwise

To step down 120 household current to 12 volts, we would need a transformer with a ratio of turns between the primary and secondary transformer of, A)1 to 1 B)10 to 1 C)12 to 1 D)100 to 1 E)120 to 1

Sine Wave   = angular frequency = number of radians per second  f = frequency = number of cycles per second  T = period = time for one cycle  1 cycle = 2  radians  f =  /2   T = 1/f = 2  /  ¼ cycle t = ¼ T  /2 rad. ½ cycle t = ½ T  rad. ¾ cycle t = ¾ T  /2 rad.

AC vs. DC  Voltage and current vary sinusoidally with time   Voltage and current will have a frequency and angular frequency    in radians per second  Capacitors and inductors can produce resistance-like effects   Circuits have natural oscillation frequencies  May get resonance

V and I in Phase

Time Dependence  The current and voltage values vary with time   But, the variation follows a known pattern  sine wave  We can discuss certain key values  Namely,   The maximum value (V max, I max )   Can think of as an average

Max Values  The value at any time is just the maximum value times the sinusoidal factor:  V = V max sin  t  I = I max sin  t   Note: Ohm’s Law still holds 

rms Values  Since the current varies in a systematic manner we would like to have some sort of average value  However the average of a sinusoidal variation is 0   Since power depends on I 2 (P =I 2 R) it does not care if the current is positive or negative

Finding rms

rms Current and Voltage  We can write the rms (root mean squared) current as: I rms = I max /(2) ½ = I max   We can write a similar relationship for the voltage  V rms =  V max /(2) ½ = V max   e.g. V max = I max R and V rms = I rms R

Resistors and AC  We can use Ohm’s law in an AC circuit with a resistor   The current and the potential difference are in phase   Large  V produces large current

AC Circuit with Resistor

Today’s PAL  Consider a AC circuit with a 240  resistance lightbulb connected to V rms = 120 V, 60 Hz household current.  For the lightbulb, what is:  the rms current  the maximum current  the maximum power  the average (rms) power  the power at time equals 1/120 second

Capacitors and AC  Consider a capacitor connected to an AC voltage source   When the current changes direction it moves the charge back and decreases the voltage  The capacitor is constantly being charged and discharged   In a AC circuit the current will vary with some average rms value that depends on the voltage and the capacitance  Capacitor acts as a resistor

AC Circuit with Capacitor

Reactance  The capacitor impedes the flow of the current  X C = 1/(  C)  The reactance, current and voltage across the capacitor are related by:  V C = IX C   At high frequency the capacitor never gets much charge on it  The voltage and the current across the capacitor are not in phase

Phase  The voltage and current across the capacitor are offset   Since the capacitor offers no resistance  As voltage increases current decreases   We say the voltage lags the current by 90 degrees  Voltage and current are offset by 1/4 cycle

AC Capacitor Phase Lag

Capacitor Power   Since P = VI, we can see if the power is positive or negative based on the sign of I and V  P is positive half the time and negative half the time   The capacitor draws energy from and returns energy to the generator in equal measure

Next Time  Read  Homework: Ch 21, P 58, 60, 61, 62