1) Consider a line of charge of length L along the x-axis spanning from x-0 to +

Question

1) Consider a line of charge of length L along the x-axis spanning from x-0 to +L. The charge is uniformly distributed over this length with total charge +Q and charge density A. We are going to work on finding the potential V at a point on the z-axis. See diagram. (25 pts) a) Write the charge density. A, in terms of dq and dx. x 0 b) Write an expression for the distance r from a small piece of the line of charge, dq, to the point P(0.0.2) in terms of x and z. c) Set up the integration to find the potential at point P(0.0.Z) with appropriate limits Solve the integral. Hints: you have help on your equation sheet and In(a)-In(b)-In(*) .Show that the potential is V k.n)

Explanation / Answer

1. given

line charge of length L

from x = 0 to x = L

charge is uniformly distributed

net charge = Q

charge densoty lambda = Q/L

a. hence

lambda = Q/L = dQ/dx

b. for small peice of charge dq at (x,0,0)

distance from (0,0,z) is

r = sqroot(x^2 + z^2)

c. hence

dV = kdq/r = k*lambda*dx/sqroot(x^2 + z^2)

d. integrating dV

V = k*lambda*ln(x + sqroot(x^2 + z^2)) + C

where C is constant of integration

now,

when x = 0, V = 0

hence

C = -k*lambda*ln(z)

hence

V = k*lambda*ln((x + sqroot(x^2 + z^2))/z)

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