PHY 111
Class #35, 4/8/16
Spring 2016
1. AC source + RLC in series
Imagine a single-loop circuit with a standard AC source and one each of R, L, and C. Using Kirchoff’s voltage rule,
we are lead to the following differential equation:
ⅆ2 I
ⅆt2
R
+L
ⅆI
ⅆt
1
+  L C  I = ω Vmax cos(ω t)
The solution is more complicated than the single or double component equations we’ve solved so far. So I’m going to
fall back onto “ICBS” (“It can be shown”) and state the solution (after exponential terms have dropped to near zero):
I(t) =
Vmax
Z
sin(ω t + φ)
where φ is a phase angle given by
tan(φ) =
XL -−XC
,
R
and Z is the combined impedance of the three elements:
Z=
R2 + XL2 -− XC2  .
The impedance is the combined “impeding” action on the current, caused by R, L, and C.
We did an example (#24 in the text) with the following given numbers: Vmax = 150 V , f = 50 Hz, R = 40 Ω,
L = 185 mH, and C = 65 μF. We are asked to calculate the maximum current in the circuit.
Solution: Maximum current is given by Imax = Vmax /∕ Z (when the sine function is = 1). So mainly we need to compute Z. Step-by step:
XL = ω L = 2 π f L = 2 π (50 Hz) 185 × 10-−3 H = 58.1 Ω
XC = 1/∕(ω C) = 1/∕(2 π f C) = 12 π (50 Hz) 65 × 10-−6 F = 49.0 Ω
Z=
R2 + XL2 -− XC2  =
(40 Ω)2 + (58.1 Ω -− 49.0 Ω)2 = 41.0 Ω.
Therefore,
Imax =
Vmax
Z
=
150 V
41.0 Ω
= 3.66 A
Done.
2. RLC resonance
PHY 111 Spring 2016
Note that Z is minimized when XL = XC , giving Zmin = R. The current reaches it largest possible value in this case,R. Martin
which is called RLC resonance. At resonance,
2
2. RLC resonance
Note that Z is minimized when XL = XC , giving Zmin = R. The current reaches it largest possible value in this case,
which is called RLC resonance. At resonance,
XL = XC so Zresonance = R,
1
ωresonance =
LC
,
(Imax )resonance =
Vmax
Zresonance
tan(φresonance ) =
(XL -−XC )resonance
R
=
Vmax
,
R
and
= 0,
We did an example of resonance in tuning in a Wifi signal at 6 MHz with an RLC circuit with L = 10-−4 H, and a
variable capacitor. The Wifi signal is an electromagnetic wave and, as we’ll see next time, has time-varying electric
and magnetic fields. IT can be “tuned in” by a Wifi receiver using an RLC resonant circuit: if the resonant frequency
is the same as the wave frequency, then the circuit will have a strong induced current (Faraday’s law) in it. Thus, if
the wave frequency, fwave = 6 MHz, is equal to the resonant frequency:
ωresonance =
1
LC
= 2 π fwave
and solving for C:
C=
1
(2 π fwave )2 L
=
1
2
2 π 600 × 106 Hz 10-−4 H
= 2.8 × 10-−10 F = 280 pF
3. Maxwell’s Equations
We ended class writing down the equations for the three major sources of electric and magnetic field:
charge causes E
current causes B
changing B flux causes E
(Coulomb’s Law, Gauss’ Law)
(Biot-Savart law, Ampere’s Law)
(Faraday’s Law)
Maxwell noticed an asymmetry in these 3 experimentally determined laws: there is a material source of both E and B,
but only E has a “field source” (time-changing magnetic flux). He predicted an extra term to allow a changing electric
flux to induce a magnetic field, restoring a “pleasing” symmetry. The next question: is it really there in the real
world? Stay tuned for the next class and find out.
PHY 111 Spring 2016
R. Martin