The force exerted by an electric charge at the origin on a charged particle at a

Question

The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector

r =

x, y, z

is F(r) = Kr/|r|3

where K is a constant. Find the work done as the particle moves along a straight line from (3, 0, 0) to (3, 2, 3).

The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = > is F(r) = Kr/|r|^3 where K is a constant. Find the work done as the particle moves along a straight line from (3, 0, 0) to (3, 2, 3).

Explanation / Answer

The straight line path is parametrized by (1 - t)(3, 0, 0) + t(3, 2, 3) = <3, 2t, 3t> with tangent vector <0, 2, 3>.
The line integral for the work done is
W = Integral of F.dr
W = Integral {t = 0 to 1} [K / (3^2 + 4t^2 + 9t^2]^3/2 <3, 2t, 3t> . <0, 2, 3>
W = Integral {t = 0 to 1} [13tK / (9 + 13t^2]^3/2 dt
let 9 + 13t^2 = u
26tdt = du
limit change for u from 9 to 22
W = Integral {u = 9 to 22} [K / 2u^3/2] du
integrating wrt u, we get
W = {u = 9 to 22} [K / 2] * -2u^-1/2]
apply limits
W = -K * (1/sqrt(22) - 1/sqrt(9)]
W = K * (1/3 - 1/sqrt(22)]
Note that the work is positive, which makes sense since the particle is moving further away from the origin, and the electric field points radially outward.

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