Chapter 12.4, Problem 73E

Electric potential function The electric potential in the xy-plane associated with two positive charges, one at (0, 1) with twice the magnitude as the charge at (0, –1), is

     $\phi \left(x,y\right)=\frac{2}{\sqrt{x^2+{\left(y-1\right)}^2}}+\frac{1}{\sqrt{x^2+{\left(y+1\right)}^2}}.$

a.    Compute ϕ_x and ϕ_y.

b.    Describe how ϕ_x and ϕ_y behave as x, y → ±∞.

c.    Evaluate ϕ_x(0, y), for all y ≠ ±1. Interpret this result.

d.    Evaluate ϕ_y(x, 0), for all x. Interpret this result.