2012 tutorial 1 with solutions

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