Theoretical Physics
Assignment 3 (Deadline: 20 Oct 2014)
1.
From Kirchhoff’s law the current I in an RC (resistance-capacitance) circuit obeys the equation
R
dI 1
 I  0.
dt C
(a) Find I(t).
(b) For a capacitance of 10,000 microfarads charged to 100 volts and discharging through a
resistance of 1 mega-ohm, find the current I for t=0 and t=100 seconds.
Note: The initial voltage is I0R or Q/C, where Q 
2.
0
I ( t )dt .
The motion of a body falling in a resisting medium may be described by
m
3.


dv
 mg  bv
dt
when the retarding force is proportional to the velocity, v. Find the velocity. Evaluate the
constant of integration by demanding that v(0)=0.
Verify that
g(  )
h(  ) 

 2 ( r , , )  k 2  f ( r )  2  2
 ( r , , )  0
r
r sin 2  

4.
is separable (in spherical polar coordinates). The function f, g and h are functions only of the
variables indicated; k2 is a constant.
Transform our linear, second-order, differential equation
y   P( x ) y   Q( x ) y  0
by the substitution y  z exp 12


x
P( t )dt  and show that the resulting differential equation

for z is
z   q( x )z  0
where q( x )  Q( x )  12 P( x )  14 P ( x ).
2
5.
Show, by means of the Wronskian, that a linear, second-order , homogeneous, differential
equation of the form
y ( x )  P( x ) y ( x )  Q( x ) y( x )  0
6.
can not have three independent solutions. (Assume a third solution and show that the Wronskian
vanishes for all x.)
Given that one solution of
1
m2



R  R  2 R0
r
r
is R  r , show that Eq. (3.56) predicts a second solution, R  r
m
m
.