Question

A very small sphere with positive charge 5.00uC is released from rest at a point 1.20cm from a very long line of uniform linear charge density 3.00 uC/m.

What is the kinetic energy of the sphere when it is 4.50cm from the line of charge if the only force on it is the force exerted by the line of charge?

Answer 1

Let us situate this on the \(x\) axis, and let our uniform line of charge be positioned on the interval \((-L, 0]\) for some large number \(L\). The voltage \(V\) as a function of \(x\) on the interval \((0,\infty)\) is given by integrating the contributions from each bit of charge. Let the charge density be \(\lambda\). Thus, for an infinitesimal length element \(dx'\), we have \(\lambda = \frac{dq}{dx'}\).\[V(x) = \frac{1}{4\pi \epsilon_0} \int\limits_{\text{line}} \frac{dq}{r}=\frac{\lambda}{4\pi \epsilon_0} \int\limits_{-L}^0 \frac{dx'}{x-x'}=\frac{\lambda}{4\pi \epsilon_0} \left( \ln|x+L| -\ln|x|\right)\]Try using this now to find an approximate voltage drop over your interval. I don't really think you can do it exactly without knowing the value of \(L\).