Energy renormalization in a simple system

If you put an electron near an infinitely long wire of uniform charge per unit length \(\lambda = 1~C/m\), the electron will be attracted toward the wire since the signs of charge are opposite. If you try to find the electrical potential energy \(U(d)\) at the electron's position d from the wire, you will find an infinite value. Correspondingly it's hard to show that's the potential energy will decrease if the electron is closer with the wire, since we're comparing infinity to infinity.

We get this infinity because we by default chose the potential energy to be zero very far from the wire. We could, however, compensate for the infinite potential energy at d by redefining the potential energy far away to be positive infinity. Amazingly, by adding a positive infinite value to a negative infinite value in just the right way, we can get a finite value for the redefined potential! This is the idea behind renormalization in quantum field theory.

Let's do this. First, let us cut off the length of the wire to be some very large L (with the electron equidistant from the ends of the wire) and set the potential energy at infinity to be zero (the usual case). Calculate the potential energy in this situation. Separate this potential energy \(U(d,L)\) into a part \(I(L)\) that does not depend on d and a part \(W(d)\) that does have d dependence, \(U(d,L)=I(L)+W(d)\). Since all quantities are finite this is a well defined problem and we'll get some finite value for W(d). We will now redefine the potential energy to be \(E(d)=U(d,L)-I(L)\).

We can take the limit of this finite problem to get our problem with a well defined E(d) by letting \(I(L)=\frac{\lambda e \ln{(L)}}{2 \pi \epsilon_o}\) in the limit \(L\rightarrow \infty\). Hence we have effectively subtracted off the divergent piece and "renormalized" our electric potential energy.

Find the value of E(d) in Joules using this procedure if the electron is 1 m from the wire.

Note that the motion of the electron can be perfectly well calculated with this potential energy as only differences in potential energy matter. Hence that constant that we pulled out is irrelevant to the physics (which is why this whole procedure works).

Details and assumptions

  • The electron's charge is \(e=-1.6 \times 10^{-19}(C)\) and the electric permittivity is \(\epsilon_o=8.85 \times 10^{-12}(F/m)\).

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