Question

As a member of a team of storm physicists, you are attempting to replicate lightning by charging two long cables stretched over a canyon, as shown. One cable will attain a highly positive (and uniform) charge density of ? and the other will attain the same amount of charge density, but opposite in sign (i.e.,

Answer 1

The flux out of a Gaussian cylinder length L radius r containing a line charge density ? is:

?E.dA = q_in/?0 = ?L/?0

Since E is constant and radial over the cylinder (by symmetry) we can remove it from the integral:

E?dA = ?L/?0

The surface area of the cylinder (excluding the ends where there is no flux) is:

?dA = 2?rL

E1 = ?/2??0r

For the negative charge density the field over a cylinder radius R is:

E2 = -?/2??0R

E at points between the two cables is given by the sum of the two fields. We may also use the fact that between the wires r + R = D to replace R in the equation for E2 giving the expression in terms of the radius from the positively charged cable:

E_tot = E1 + E2 = ?/2??0r - ?/2??0(D - r)

E_tot = (?/2??0)( (1/r) - (1/(D - r)) )