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18
MIDSUMMER EXAMINATIONS 2002
Subject
MATHEMATICAL PHYSICS
Title of Paper
MODULE PA174
Time Allowed
Three hours
MATHEMATICAL PHYSICS 2A
____________________________________________________________________________________________________________
Instructions to candidates
Candidates may answer any number of questions from Sections A and B.
Section A is worth about 40 marks in total. Questions in Section B carry 20 marks each.
A mark of 70% corresponds to a total of 70 marks gained.
SEPARATE ANSWER BOOKS MUST BE USED FOR SECTIONS A AND B.
____________________________________________________________________________________________________________
SECTION A
sin(2x + y)
.
x+y
A1.
Verify that fxy = fyx when f (x, y) =
A2.
Find the directional derivative of f(x,y) = 3x2 + y3 at the point(1,1) in the direction that
make an angle of π/6 wih the x-axis.
A3.
Evaluate
A4.
Find the real and imaginary parts of
∫∫
1
1
0
0
y cos(π xy)dxdy.
1+ 2i
.
2 − 3i
/continued …
2.
A5.
a)
Find the modulus and argument of 3 + 4i.
b)
Express 2exp(4πi/3) in the form x + iy and indicate its position in the Argand
diagram.
A6.
Find the three cube roots of i.
A7.
By manipulating complex exponentials show that cos2θ = cos2 θ − sin2 θ .
A8.
Find the general solution of the differential equation
y ′′ − 2y ′ − 3y = 0 .
A9.
Find a particular integral of the equation
y ′′ + y ′ + y = x 2 + 1.
A10.
Write down expressions for the complex impedance of the following elements driven
by an AC source of frequency ω.
a)
b)
c)
d)
A11.
A pure resistor R.
A pure inductor L.
A pure capacitor C.
A resistor R in series with a parallel combination of an inductor L and a
capacitor C.
A 10 A peak-to-peak current passes through a 10 pF capacitor that is in series with a
40 Ω resistor. What is the power dissipation?
/continued …
3.
A12.
Write down the expression for the real impedance of a series circuit containing an
inductor L, a capacitor C and a resistor R driven by an AC source of frequency ω. If
the component values are L = 0.05 H, C = 1 µF and R = 50 Ω, at what frequency will
current resonance occur and what is the value of the Q-factor?
SECTION B
B1.
B2.
a)
A building has a hemispherical roof of radius a. On a certain day half of the
roof is illuminated by the setting sun. The solar power per unit area incident on
the roof is ksinθ cosφ kW m–2 where θ and φ respectively are polar and
aximuthal angle in a co-ordinate system where the z-axis is vertical and the xaxis points to the sun. What is the total solar power incident on the roof?
b)
A roll of kitchen towels of outer radius a, inner radius b and height h is
splashed by water from a sink. The density of water absorbed in the roll is
k
−z/l
( x + a)e k gm–3 in a co-ordinate system where the origin is on the axis of
2a
the roll at its base. What is the total mass of water absorbed in the roll?
c)
The charge density in a spherical piece of insulator of radius a is kz2 Cm–3 in a
co-ordinate system where the origin is at the centre of the sphere. What is the
total charge stored in the sphere?
Find the general solutions of the following differential equations
i)
x 3 y ′ = e −y
ii)
xy ′ + 2y = 4x 2
iii)
y ′′ − 2y ′ + y = 0
iv)
y ′′ + 2 y ′ − 3y = 8e −x .
/continued …
4.
B3.
Show that 1 is a root of
z3 – 4z2 + (6 + i)z – (3 + i) = 0,
and hence find the other two roots. Explain why this is an example of the fundamental
theorem of algebra.
B4.
i)
Find the general solution of the following differential equation
y ′′ + y ′ − 2y = 3e x .
Find the solution that satisfies the initial conditions y(0 ) = 0 and y ′(0) = 0 .
ii)
Find the general solution of the following differential equation
y ′′ − 6y ′ + 10 y = 0 .
Express your solution in terms of real functions of the variable x.
B5.
A circuit is constructed with an uncharged capacitor C in series with a resistor R, a
switch S and a d.c. voltage source of value V0. At time t = 0, the switch is closed.
a)
Write down equations for the voltage drop across the resistor R and the
capacitor C.
b)
Derive an expression for the charge on the capacitor as a function of time.
Sketch a graph to show this function.
c)
What is the response of the circuit (current as a function of time)? Sketch a
graph of this function.
d)
At a time t = 2RC, calculate the charge on the capacitor as a fraction of the
maximum it can store.
e)
The d.c. voltage source is replaced by one giving a square wave with a
minimum voltage of 0 V, a maximum voltage of Vmax and a period of 200 ms.
If the switch remains closed at all times and the circuit has a time constant of
RC = 100 ms, sketch how the charge stored on the capacitor varies with time.
/continued …
5.
B6.
a)
Write down the equations that are used to calculate the total complex
impedance of a combination of components in series and in parallel.
Z1
Z2
D
Z4
Z3
V
A general AC Bridge circuit is shown in the figure above, where Z1, Z2, Z3 and
Z4 represent the complex impedance in each branch of the bridge, D is an AC
meter and V is an AC voltage source.
b)
By considering the potential drops around the circuit, show that the equation
for the bridge to balance is given by Z1Z3 = Z2Z4.
c)
Explain why two conditions must be satisfied for the bridge to balance.
A Schering bridge can be formed by putting the following components into
each branch of the bridge. Z1 is a capacitor C1 in series with a resistor R1. Z2 is
a resistor R2. Z3 is a capacitor C3 in parallel with a resistor R3. Z4 is a capacitor
C4.
d)
Write down the total complex impedance in each branch of the bridge.
e)
Determine the balance conditions for the Schering bridge.
f)
If the Schering bridge components have the following values; R1 = 100 Ω,
C1 = 1 µF, R2 = 1 kΩ and C4 = 10 nF, then what values of the capacitor C3 and
resistor R3 are required to make the bridge balance?