Description
Problem 1. One application of LRC series circuits is to high-pass or low-pass filters, which filter out
either the low- or high-frequency components of a signal.
(a) In a high-pass filter the output voltage is taken across the LR combination (see the
figure below). Derive an expression for Vhi/Vs, the ratio of the output and source
amplitudes as a function of the angular frequency ω of the source. Show that when
ω is small, this ratio is proportional to ω and thus is small, and show that the ratio
approaches unity in the limit of large frequency.
(b) In a low-pass filter the output voltage is taken across the capacitor in an LRC circuit.
Derive an expression for Vlo/Vs , the ratio of the output and source amplitudes as a
function of the angular frequency ω of the source. Show that when ω is large this
ratio is proportional to ω
−2 and thus is small, and show that the ratio approaches
unity in the limit of small frequency.
(3 + 3 marks)
Problem 2. A resistor, inductor, and capacitor are connected in parallel to an AC source with voltage
amplitude V and angular frequency ω. Let the source voltage be given by v(t) = V cos ωt.
(a) Argue that the instanteneous voltages vR, vL, and vC at any instant are each equal
to v and that i = iR + iL + iC, where i is the current through the source and iR,
iL, and iC, are the currents through the resistor, the inductor, and the capacitor,
respectively.
(b) What are the phases of iR, iL, and iC with respect to v? Draw the corresponding
diagram on the complex plane.
(c) Show that the current amplitude I for the current i throught the source is given
by I =
q
I
2
R + (IC − IL)
2 and this result can be written as I = V /Z, with Z
−1 =
p
1/R2 + (ωC − 1/ωL)
2.
(d) Show that at the angular frequency ω0 = 1/
√
LC, IC = IL and I is a minimum. Since
I is a minimum at resonance, is it to correct to say that the power delivered to the
resistor is also a minimum at ω = ω0? Explain.
(e) At resonance, what is the phase angle of the source current with respect to source
voltage? How does this compare to the phase angle for an LRC series circuit at
resonance?
(1 + 1 + 2 + 1 + 1 marks)
Problem 3. Electromagnetic radiation is emitted by accelerating charges. As we briefly mentioned in
class, the rate at which energy E is emitted by an accelerating charge q is given by the
Larmor formula (in SI units) dE
dt =
q
2a
2
6πε0c
3 , where a is acceleration of the charge and c is
the speed of light.
(a) If a proton with a kinetic energy of 6 MeV is traveling in a particle accelerator in a
circular orbit of radius 0.75 m, what fraction of its energy does it radiate per second?
(b) Consider an electron orbiting with the same speed and radius. What fraction of its
energy does it radiate per second?
(c) (the ”classical hydrogen atom”) Consider the electron in a hydrogen atom as a particle
moving in a circular orbit of radius 0.0539 nm, with kinetic energy of 13.6 eV. If the
electron behaved classically, how much energy would it radiate per second? What
does it tell you about the use of classical physics in describing the atom within this
model?
(1 + 1 + 2 marks)
Problem 4. Let f1(x, t) = Ae−k(x−vt)
2
, f2(x, t) = A sin [k(x − vt)], f3(x, t) = A
k(x−vt)
2+1 , and f4(x, t) =
Ae−k(kx2+vt)
, f5(x, t) = A sin(kx) cos(kvt)
3
. Check that the functions f1, f2, and f3 satisfy
the 1D classical wave equation, but f4 and f5 do not.
(5 × 1 marks)
Problem 5. Show that (a) the standing wave ξ(x, t) = A sin(kx) cos(ωt) satisfies the wave equation and
(b) express it as a sum of a wave traveling to the left and a wave traveling to the right.
(1 + 1 marks)
Problem 6. Rewrite the classical wave equation ∂
2
ξ(x,t)
∂x2 −
1
v
2
∂
2
ξ(x,t)
∂t2 = 0 using new variables α = x+vt,
β = x−vt and show that any solution of this equation may be expressed as a sum of a wave
traveling to the left and a wave traveling to the right, i.e. ξ(x, t) = ξ1(x + vt) + ξ2(x − vt).
(2 marks)
Problem 7. Electromagnetic waves propagate much differently in conductors than they do in dielectrics
or in vacuum. If the resistivity of the conductor is sufficiently low (that is, if it is a sufficiently good conductor), the oscillating electric field of the wave gives rise to an oscillating
conduction current that is much larger than the displacement current. In this case, the
wave equation for an electric field E(x, t) = (0, Ey(x, t), 0) propagating in the +x direction
within a conductor is
∂
2Ey(x, t)
∂x2
=
µ
ρ
∂Ey(x, t)
∂t ,
where µ is the permeability of the conductor and ρ is its resistivity.
(a) Check that
Ey(x, t) = E0e
−kCx
sin(kCx − ωt),
where kC =
p
ωµ/2ρ, is a solution of the above wave equation.
(b) The exponential term shows that the electric field decreases in amplitude as it propagates. Explain why this happens.
Hint. The field does work to move charges within the conductor. The current of these
moving charges causes heating within conductor. Where does this energy come from?
(c) Note that the electric-field amplitude decreases by a factor of 1/e in a distance 1/kC
and calculate this distance for a radio wave with frequency ν = 1 MHz in copper
(resistivity 1.72 × 10−8 Ω · m, relative permeability µr = 1). Since the distance is so
short, electromagnetic waves of this frequency can hardly propagate at all into copper.
Instead they are reflected at the surface of the metal. This is why radio waves cannot
penetrate through copper or other metals.
(2 + 3 + 1 marks)
