Physics 213 General Physics Lecture 13

1 Last Meeting: Self Inductance, RL Circuits, Energy Stored Today: Finish RL Circuits and Energy Stored. Electric Generators, Alternating Current

Inductor in a Circuit Inductance can be interpreted as a measure of opposition to the rate of change in the current  Remember resistance R is a measure of opposition to the current As a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing current  Therefore, the current doesn’t change from 0 to its maximum instantaneously

RL Circuit When the current reaches its maximum, the rate of change and the back emf are zero The time constant, , for an RL circuit is the time required for the current in the circuit to reach 63.2% of its final value

Energy Stored in a Magnetic Field The emf induced by an inductor prevents a battery from establishing an instantaneous current in a circuit The battery has to do work to produce a current  This work can be thought of as energy stored by the inductor in its magnetic field  PE L = ½ L I 2

A long, straight wire is in the same plane as a rectangular, conducting loop.The wire carries a constant current I as shown in the figure. Which one of the following statements is true if the wire is suddenly moved toward the loop? (a) There will be no induced emf and no induced current. (b) There will be an induced emf, but no induced current. (c) There will be an induced current that is clockwise around the loop. (d) There will be an induced current that is counterclockwise around the loop. (e) There will be an induced electric field that is clockwise around the loop. X

A long, straight wire is in the same plane as a rectangular, conducting loop.The wire carries a constant current I as shown in the figure. Which one of the following statements is true if the wire is suddenly moved toward the loop? (a) There will be no induced emf and no induced current. (b) There will be an induced emf, but no induced current. (c) There will be an induced current that is clockwise around the loop. (d) There will be an induced current that is counterclockwise around the loop. (e) There will be an induced electric field that is clockwise around the loop. X

AC Circuits An AC circuit consists of a combination of circuit elements and an AC generator or source The output of an AC generator is sinusoidal and varies with time according to the following equation  Δv = ΔV max sin 2  ƒt Δv is the instantaneous voltage ΔV max is the maximum voltage of the generator ƒ is the frequency at which the voltage changes, in Hz

AC Generators The emf generated by the rotating loop can be found by ε =2 B ℓ v  =2 B ℓvsin θ If the loop rotates with a constant angular speed, ω, and N turns ε = N B A ω sin ω t ε = ε max when loop is parallel to the field ε = 0 when when the loop is perpendicular to the field

Resistor in an AC Circuit Consider a circuit consisting of an AC source and a resistor The graph shows the current through and the voltage across the resistor The current and the voltage reach their maximum values at the same time The current and the voltage are said to be in phase

Dissipation Across Resistors in an AC Circuit The rate at which electrical energy is dissipated in the circuit is given by P=i 2 R=v 2 /R Where I and v are the instantaneous current and voltage across resistor The maximum current occurs for a small amount of time Average current is zero. Average power > zero.

rms Current and Voltage The rms current is the direct current that would dissipate the same amount of energy in a resistor as is actually dissipated by the AC current Alternating voltages can also be discussed in terms of rms values

Power Revisited The average power dissipated in resistor in an AC circuit carrying a current I is 

Ohm’s Law in an AC Circuit rms values will be used when discussing AC currents and voltages  Many of the equations will be in the same form as in DC circuits Ohm’s Law for a resistor, R, in an AC circuit  ΔV R,rms = I rms R Also applies to the maximum values of v and i

Capacitors in an AC Circuit The current reverses direction The voltage across the plates decreases as the plates lose the charge they had accumulated The voltage across the capacitor lags behind the current by 90°

Capacitive Reactance and Ohm’s Law capacitive reactance  When ƒ is in Hz and C is in F, X C will be in ohms Ohm’s Law for a capacitor in an AC circuit  ΔV C,rms = I rms X C

Inductors in an AC Circuit Consider an AC circuit with a source and an inductor The current in the circuit is impeded by the back emf of the inductor The voltage across the inductor always leads the current by 90°  v = L  I/  t

Inductive Reactance and Ohm’s Law inductive reactance X L = 2  ƒL When ƒ is in Hz and L is in H, X L will be in ohms Ohm’s Law for the inductor  ΔV L,rms = I rms X L

Combined Circuits: The RLC Series Circuit The resistor, inductor, and capacitor can be combined in a circuit The current in the circuit is the same at any time and varies sinusoidally with time

Current and Voltage Relationships in an RLC Circuit The instantaneous voltage across the resistor is in phase with the current The instantaneous voltage across the inductor leads the current by 90° The instantaneous voltage across the capacitor lags the current by 90°

Resonance in an AC Circuit Resonance occurs at the frequency, ƒ o, where the current has its maximum value  To achieve maximum current, the impedance must have a minimum value  This occurs when X L = X C  Then,