P14B Tutorial Sheet 1 Integration and Differentiation For questions 1-3 below assume dr dV E . 1. Show that if 2 r kq E then r kq V r , 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 r R kq E 3 then ) ( 2 2 2 3 r R R kq V V R r , assuming that (i) r can vary between R (lower limit) and r (upper limit) (ii) k is a constant. 3. Show that if Lr kQ E 2 then a b L kQ V V b a 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 kq V , then 2 r kq E . Powers, Brackets and Binomial Expansion 5. Show that ) 3 1 ( 1 ) ( 3 3 r a r a r and that ) 3 1 ( 1 ) ( 3 3 r a r a r when r >> a. Hence show that 4 3 3 6 ) ( ) ( r a a r a r . Combining everything from class 6. For the diagram s, a and r are lengths: a. Show that if both E 1 and E 2 (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 kqa E , where the magnitudes of E 1 and E 2 are the same and equal to 2 s kq , and k and q are constant. b. Show also that this can be rewritten as: 2 3 2 2 ) ( 2 a r kqa E . c. Show that if r >> a, the expression can be simplified as 3 2 r kqa E . E 1 E 2 s r a a