Question

6. Consider a thin rod of length L laying on the x-axis as shown in the figure below. The charge density is non uniform and varies as λ = λ。x/L, where λ, is a constant. (a) Find the magnitude of the force on a point charge q at x-a in terms of q, λ, a, and L. (b) Show that for a » L the answer in (a) reduces to (c) What is the total charge on the rod? (d) Does your answer in c make sense in light of the result given in (b)? Why or why not? X E -L x=0 x=a

Answer 1

Given that A thin rod length L laying on x-axis here dQ = lambda dx dQ = lambda_0 x/L dx F_p = ? dF_p = k d Qq/x^2 F_p = k integral^a + 1_a lambda_0 x q/L x^2 dx = k lambda_0 q/L [ln x]^a + L_a F_p = k lambda_0 q/L ln (a + L/a) b) F_p = k lambda_0 q/L ln (1 + L/a) for L < < a ln(1 + L/a) = L^2/2a^2 so, F_p = k lambda_0 q/L L^2/2a = k lambda_0 q_L/2a^2 c) total charge on rod integral dQ = integral^L_0 lambda_0/L x dx Q = lambda_0/L [x^2/2]^L_0 Q = lambda_0 L/2