# 110/16/2015 Applied Physics Lecture 19  Electricity and Magnetism Induced voltages and induction Energy AC circuits and EM waves Resistors in an AC circuits

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• **Applied PhysicsLecture 19 Electricity and Magnetism

Induced voltages and inductionEnergyAC circuits and EM wavesResistors in an AC circuits

• **Homework AssignmentDue next class 19.7,11,3320.1,7,9,24,28,37

• **Inductor in a CircuitInductance can be interpreted as a measure of opposition to the rate of change in the currentRemember 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 currentTherefore, the current doesnt change from 0 to its maximum instantaneouslyMaximum current:

• **20.9 Energy stored in a magnetic fieldThe battery in any circuit that contains a coil has to do work to produce a currentSimilar to the capacitor, any coil (or inductor) would store potential energy

Summary of the properties of circuit elements.ResistorCapacitorInductorunitsohm, W = V / Afarad, F = C / Vhenry, H = V s / AsymbolRCLrelationV = I RQ = C Vemf = -L (DI / Dt)power dissipatedP = I V = I R = V / R00energy stored0PEC = C V / 2PEL = L I / 2

• **Example: stored energyA 24V battery is connected in series with a resistor and an inductor, where R = 8.0W and L = 4.0H. Find the energy stored in the inductor when the current reaches its maximum value.

• **A 24V battery is connected in series with a resistor and an inductor, where R = 8.0W and L = 4.0H. Find the energy stored in the inductor when the current reaches its maximum value.Given:

V = 24 VR = 8.0 WL = 4.0 H

Find:

PEL =?Recall that the energy stored in th inductor isThe only thing that is unknown in the equation above is current. The maximum value for the current isInserting this into the above expression for the energy gives

• Chapter 21Alternating Current Circuits and Electromagnetic Waves

• **AC CircuitAn AC circuit consists of a combination of circuit elements and an AC generator or sourceThe output of an AC generator is sinusoidal and varies with time according to the following equation

V = Vmax sin 2t

v is the instantaneous voltageVmax is the maximum voltage of the generator is the frequency at which the voltage changes, in Hz

Same thing about the current (if only a resistor)

I = Imax sin 2t

• **Resistor in an AC CircuitConsider a circuit consisting of an AC source and a resistorThe graph shows the current through and the voltage across the resistorThe current and the voltage reach their maximum values at the same timeThe current and the voltage are said to be in phase

Voltage varies as

V = Vmax sin 2t

Same thing about the currentI = Imax sin 2t

• **More About Resistors in an AC CircuitThe direction of the current has no effect on the behavior of the resistorThe rate at which electrical energy is dissipated in the circuit is given by

P = i2 R = (Imax sin 2t)2 R

where i is the instantaneous currentthe heating effect produced by an AC current with a maximum value of Imax is not the same as that of a DC current of the same valueThe maximum current occurs for a small amount of time

Averaging the above formula over one cycle we get

• **rms Current and VoltageThe 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

• **Ohms Law in an AC Circuitrms values will be used when discussing AC currents and voltagesAC ammeters and voltmeters are designed to read rms valuesMany of the equations will be in the same form as in DC circuitsOhms Law for a resistor, R, in an AC circuit

Vrms = Irms R

Also applies to the maximum values of v and i

• **Example: an AC circuitAn ac voltage source has an output of DV = 150 sin (377 t). Find (a) the rms voltage output, (b) the frequency of the source, and (c) the voltage at t = (1/120)s. (d) Find the maximum current in the circuit when the generator is connected to a 50.0W resistor.

• **Capacitors in an AC CircuitConsider a circuit containing a capacitor and an AC sourceThe current starts out at a large value and charges the plates of the capacitorThere is initially no resistance to hinder the flow of the current while the plates are not chargedAs the charge on the plates increases, the voltage across the plates increases and the current flowing in the circuit decreases

• **More About Capacitors in an AC CircuitThe current reverses directionThe voltage across the plates decreases as the plates lose the charge they had accumulatedThe voltage across the capacitor lags behind the current by 90

• **Capacitive Reactance and Ohms LawThe impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by

When is in Hz and C is in F, XC will be in ohmsOhms Law for a capacitor in an AC circuitVrms = Irms XC

• **Inductors in an AC CircuitConsider an AC circuit with a source and an inductorThe current in the circuit is impeded by the back emf of the inductorThe voltage across the inductor always leads the current by 90

• **Inductive Reactance and Ohms LawThe effective resistance of a coil in an AC circuit is called its inductive reactance and is given byXL = 2LWhen is in Hz and L is in H, XL will be in ohmsOhms Law for the inductorVrms = Irms XL

• **Example: AC circuit with capacitors and inductorsA 2.40mF capacitor is connected across an alternating voltage with an rms value of 9.00V. The rms current in the capacitor is 25.0mA. (a) What is the source frequency? (b) If the capacitor is replaced by an ideal coil with an inductance of 0.160H, what is the rms current in the coil?