2012 tutorial 1 with solutions
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P14B
Tutorial Sheet 1
Integration and Differentiation
For questions 1-3 below assume dr
dVE .
1. Show that if 2r
kqE then
r
kqVr , assuming that (i) r can vary between infinity (lower limit)
and r (upper limit) (ii) k is a constant and (iii) V at infinity, V is zero.
2. Show that if rR
kqE
3 then )(
2
22
3rR
R
kqVV Rr , assuming that (i) r can vary between R
(lower limit) and r (upper limit) (ii) k is a constant.
3. Show that if Lr
kQE
2 then
a
b
L
kQVV ba ln
2, assuming that (i) r can vary between a (lower
limit) and b (upper limit) (ii) k, Q and L are constants.
4. Show that if r
kqV , then
2r
kqE .
Powers, Brackets and Binomial Expansion
5. Show that )3
1(1
)(3
3
r
a
rar and that )
31(
1)(
3
3
r
a
rar when r >> a.
Hence show that 4
33 6)()(
r
aarar .
Combining everything from class
6. For the diagram s, a and r are lengths:
a. Show that if both E1 and E2 (remember E is electric field
strength) act as shown at the apex of the isosceles triangle
then the net E is given by 3
2
s
kqaE , where the
magnitudes of E1 and E2 are the same and equal to 2s
kq, and
k and q are constant.
b. Show also that this can be rewritten as:
2
3
22 )(
2
ar
kqaE .
c. Show that if r >> a, the expression can be simplified as
3
2
r
kqaE .
E1
E2
s
r
a a
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