Built on Facts

There’s a test problem famous by virtue of the fact that one student failed to answer it, but did so in a hilarious way. You’ve probably seen it, and here it is:

Let me quote the problem in case it’s hard to read.

A proton approaches a long line of positive charge so that with its initial trajectory it would intersect the line. The line has uniform charge density of 5 nC/m. If the proton starts off with a velocity 300 km/s a distance 1km from the line charge, what is the distance of closest approach?

Well it’s sloppily worded and it uses “it’s” where “its” is required. But the real problem is that it’s unclear exactly with what trajectory the proton approaches the wire. It makes a difference: a head on approach will bring it closer than than one that starts at an angle. Why? Well, the field points radially outward so the proton will feel a force radially outward. That will not change the proton’s velocity component parallel to the wire and thus there is a kinetic energy associated with that motion which will no be affected by the field. But it’s assume the proton is approaching the wire head-on, as that’s probably what the test maker intended. The test-maker’s hint suggests an attack using potential, which is a good place to start. To find the potential (if we don’t know it from a formula sheet), we can use the electric field.

The field of the wire is

Normally we’d set the zero of potential at infinity, but unfortunately the integral of a 1/r force diverges. Therefore we’ll just find the antiderivitive and evaluate it separately at each location of interest. We have (integrating over r)

Which gives

Piece of cake. Now let’s do the physics. We know the proton initially has a kinetic energy K = (1/2)mv², and a potential energy given by q times the potential V given above. So we need to find the distance r at which the potential energy of the proton is greater by K than when it started - in other words, when all kinetic energy has been lost to potential energy. Using i and f to denote initial and final distances, we have the following:

Not pretty, but it’s complete and all that’s required is to solve for r_f. We know everything else. We get the following wild-looking but true equation

So given what’s in the problem, what’s the numerical answer? Plugging in, I get a value of 5.367 meters. Now you too can find the identity of Batman and get the solution to the problem!