Question

dV dr For questions 1-3 below assume E 1Sowtat t kq assuming that () can vany between nfinity (lower limit) and r (upper limit) (ii) k is a constant and (ii) V at infinity is zero. 2. Show that if E=Rs r then V, VR= q (R2 r2), assuming that (i)rcan vary between R thatif E - kgr then V, -v, - ko (R'-), assuming that () r can vary between R 2R3 (lower limit) and r (upper limit) (i) k is a constant. 3. Show that if Ethen V. -v, - R n2, assuming that ()r can vary between a (lower Lr limit) and b (upper limit) (ii) k, Q and L are constants. A. Show that if V -kg then E-K

Answer 1

E = -dv/dr = > dv = -E dr = -kq/r^2 dr = > integral^v_0 dv = -integral^r_infinity kq/r^2 dr = -kq integral^r_infinity dr/r^2 = > v = -kq [-1/r]^r_infinity = -kq [-1/r + 1/ infinity] = -kq [-1/r] = kq/r = > v = kq/r E = kq/R^2 r integral^vr_v_R dv = integral^r_R kq/R^2 r = -kq/R^2 integral^r_R r dr = -kq/R^3 r dr = -kq/R^3 [r5^2]^r_R = > v_r - v_R = -kq/R^3 [r^2 - R^2] = kq/R^3 [R^2 - r^2] = > V_r - V_R = kq/R^3 [R^3 - r^2]