Lecture 21, Oct. 25, 2000

Recall from last lecture:

The summary table of ac properties of circuit elements.
Phasor diagrams: used for finding the voltage-current relation in an ac circuit.

Voltage: V = Sqrt{V_R² + (V_L - V_C)²}
Impedance: Z = Sqrt{R² + (X_L - X_C)²}
Phase: tanf = (V_L - V_C) / V_R = (X_L - X_C) / R

21.5 Power in an ac Circuit

As mentioned previously (see the summary table at the beginning of the lecture of Oct. 20, 2000), capacitors and inductors store energy, but do not dissipate energy. Only resistors dissipate energy. Therefore, the power dissipated in an ac circuit depends only on the resistance in the circuit

P_av = I² R.

Since R = DV_R / I, the average power can also be written:

P_av = I DV_R.

Finally, we can express DV_R = DV cosf, as can be seen from the voltage triangle of the previous section. This gives us a third form for the average power:

P_av = I DV cosf. The cos f factor is known as the power factor.

Example: P21.26

In a certain RLC circuit, the rms current is 6.0A, the rms voltage is 240V, and the current leads the voltage by 53°. (a) What is the total resistance of the circuit? (b) Calculate the total reactance, X_L - X_C. (c) Find the average power dissipated in the circuit.

Begin by drawing a phasor diagram of the rms voltage. Because the current leads the voltage, the voltage lags the current, so the phase angle is f = -53°. (a) V_R is the component of the voltage along the x-axis, and is related to the resistance by R = V_R / I_r.
V_R = DV cosf = (240V) cos-53° = 144V
R = V_R / I_r = (144V) / (6.0A) = 24W

(b)X_L - X_C = R tanf = 24W tan-53° = -32W

21.6 Resonance in a Series RLC Circuit

Resonance is a phenomenon where system responds strongly to an exciting force with a particular frequency. For instance, the string of a guitar or violin vibrates at a particular frequency when plucked, the air in a wind instrument when excited, the tines of a tuning fork, or the crystal used to keep time in an electronic (quartz) watch. Resonance also occurs in RLC circuits.

If the ac voltage source has a peak value V_m, then the peak current can be written:

I_m = V_m / Z = V_m / {R² + (X_L - X_C)²}

Remember that X_L and X_C depend on the frequency of the source. We will get a maximum current in the circuit when X_L = X_C, so that Z = R. This occurs at the resonance frequency, f₀, where

2p f₀ L = 1 / 2p f₀ C

From this we find

f₀ = 1 / 2p Sqrt{LC}.

Show a plot of the current versus frequency, as in Figure 21.23.

Note that if the resistance of the circuit, R, is zero, then the current becomes infinite at resonance. The small resistance in real devices keeps this from happening.

Applications of resonance include receivers for television and radio, and the detector in an MRI. In fact, the R in MRI stands for resonance -- magnetic resonance imaging.

Example: P21.29

An RLC circuit is used to tune a radio to an FM station broadcasting at 88.9MHz. The resistance in the circuit is 12W and the capacitance is 1.40pF. What inductance should be present in the circuit?

The resistance given is not important and will not be used. We want the resonance frequency of the circuit to equal the frequency of the station, and this depends only on the inductance and capacitance:
f₀ = 1 / 2p Sqrt{LC}
Therefore, solving for L:
L = 1 / 4p² f₀² C = 1 / 4p² (88.9×10⁶Hz)² (1.40×10^-12F) = 2.29×10^-6H = 2.29mH.

21.11 Properties of Electromagnetic Waves

I'll begin this section with some introduction from sections 21.8, 21.9, and 21.10. Since electric and magnetic fields are very similar in behavior, and, as we've seen, changing magnetic fields produce electric fields (seen by us as emf's), one might wonder if changing electric fields produce magnetic fields. In 1865, Maxwell proposed just this. When he included this effect among the equations of electricity and magnetism, he noticed that the resulting equations also allowed waves to exist, and that the speed of these waves predicted by the equations was very close to the speed of light! (speed of light = c = 3×10⁸ m/s) He concluded, and it was later verified, that light is an electromagnetic wave, that is a wave made by fluctuating electric and magnetic fields.

Experiments were carried out by Hertz that confirmed the existence of electromagnetic waves, and demonstrated that his waves (f = 100MHz) had properties very similar to light waves (f = 500THz). These properties include a propagation speed equal to the speed of light, reflection, refraction, interference, and polarization. In Maxwell's theory, the speed of electromagnetic waves is

c = 1 / Sqrt{m₀ e₀}

If we substitute the values m₀ = 4p×10^-7 N s² / C² and e₀ = 8.85×10^-12 C² / N m² we find c = 2.998×10⁸ m/s. Compare this with the exact value for c as defined by international agreement: c = 2.99792458×10⁸ m/s (exactly).

Maxwell's theory also gives us the structure of electromagnetic waves. Electromagnetic waves consist of fluctuating electric and magnetic fields, with both fields perpendicular to the direction of motion and perpendicular to each other. Because the electric and magnetic fields are perpendicular (transverse) to the motion, electromagnetic waves are transverse waves. The ratio of the magnitude of the electric and magnetic fields equals the speed of light:

E / B = c.

Electromagnetic waves carry energy and momentum! The average power per unit area transferred by an electromagnetic wave is

P_av / A = E_m B_m / 2m₀ = E_m² / 2m₀c = c B_m² / 2 m₀

The various forms are equivalent due to the relation E / B = c. The amount of momentum transferred to an absorbing surface is related to the energy U incident on the surface:

p = U / c.

If the surface is perfectly reflecting, then the momentum transfer is double the above value:

p = 2U / c.

21.12 The Spectrum of Electromagnetic Waves

Radio waves, microwaves, infrared waves, visible light, ultraviolet light, x-rays, and gamma rays are all electromagnetic waves. They differ only in their frequency, f, or wavelength, l. Because all electromagnetic waves travel (through vacuum) at the speed of light, c, the frequency and wavelength are related through:

c = f l

Figure 21.23 shows the frequency and wavelength for different classes of electromagnetic waves.

It is traditional to use frequency for waves with wavelength greater than about 1 micron. These are the microwaves and radiowaves.

For waves with wavelengths less than 1 micron, it is more common to specify the wavelength. For instance, the wavelength of visible light lies in the range from 700nm (red) to 400nm (blue).