A thin non-uniformly charged rod with length L is shown in the figure below. The

Question

A thin non-uniformly charged rod with length L is shown in the figure below. The charge linear density is lambda = ax where a is a positive constant and has units C/m^2. Point P is located on the negative x axis at a distance d from the end of the rod (P has x-coordinates = -d). Express all of your answer symbolically in terms of L, d, a, and any physical constant, as needed. a) Write down an expression for the infinitesimal charge dq. b) What is the total charge on the rod. c) Write down the distance |r vector| from the charge dq to point P. d) Write down the electric potential dV due to the infinitesimal charge dq if V = 0 at infinity. (Don't integrate yet!) e) Find the electric potential at point P due to the whole rod. (Possibly useful integral: Integral x/x + a dx = x - a In (x + a))

Explanation / Answer

a] dq = lambda*dx = alpha x dx

b] q = integral dq = alpha*x^2/2 till L

= alpha*L^2/2

c] r = d+x

d] dV = kdq/r

= k*alpha x dx /(d+x)

e] V = integral dV

= k*alpha* [x - d ln(d+x)] , now after apllying limit of integration

=  k*alpha* [L - d ln((d+L)/d)]

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