Purcell Problem 2.34

Physics · electromagnetism

Problem

Extremum of φ\varphi A charge of 2 C is located at the origin. Two charges of −1 C each are located at the points (1,1,0) and (−1,1,0). If the potential φ\varphi is taken to be zero at infinity (as usual), then it is easy to see that φ\varphi is also zero at the point (0,1,0). It follows that somewhere on the y axis beyond (0,1,0) the function φ\varphi(0,y,0) must have a minimum or a maximum. At that point the electric field E must be zero. Why? Locate the point, at least approximately.
Intuitive explanation

The key insight is that an extremum of the potential along a line must occur where the electric field perpendicular to that line vanishes. Since φ\varphi is zero at both infinity and at (0,1,0), and the potential is continuous, it must reach an extremum somewhere between them on the y-axis. At that extremum, dφdy=0\frac{d\varphi}{dy} = 0, which means the y-component of E⃗\vec{E} (the only relevant component by symmetry) must be zero—so E⃗=0\vec{E} = 0 there. Use superposition to find where the three charges' field contributions cancel along the y-axis.

Intuitive explanation

The key insight is that an extremum of the potential along a line must occur where the electric field perpendicular to that line vanishes. Since φ\varphi is zero at both infinity and at (0,1,0), and the potential is continuous, it must reach an extremum somewhere between them on the y-axis. At that extremum, dφdy=0\frac{d\varphi}{dy} = 0, which means the y-component of E⃗\vec{E} (the only relevant component by symmetry) must be zero—so E⃗=0\vec{E} = 0 there. Use superposition to find where the three charges' field contributions cancel along the y-axis.

Browse the problem library