Draw the equipotentials

Problem

Draw the equipotentials. A charged spherical conductor of radius $R$ is placed in a vacuum. Based on sections 14.5 and 14.6 of the chapter, draw on the same plane: (a) at least four electric field lines in the region $r>R$; (b) at least three equipotential surfaces; (c) indicate with arrows the direction of $\vec{E}$ and graphically verify the property $\vec{E}\perp$ equipotentials. (d) What changes if the conductor is replaced by a dipole? Make a second sketch and compare it, drawing inspiration from Epstein’s graphical constructions (Epstein 2002).

Solution

(a) Field lines of the sphere. Outside a charged conducting sphere the field is identical to that of a point charge placed at the centre: the field lines are straight rays radiating outward from the surface (if the charge is positive) and thinning out with distance, with $E \propto 1/r^2$. Draw at least four, equally spaced angularly.

(b) Equipotential surfaces. The potential depends only on $r$: $V(r) = kQ/r$. The surfaces of constant potential are therefore concentric spheres (in the plane of the drawing: concentric circles) centred on the sphere. Draw at least three at increasing radii; since $V\propto 1/r$, for equal potential steps the circles become progressively more spaced out.

(c) Orthogonality. The $\vec{E}$ arrows point radially outward. At every point the (radial) field line meets the (concentric) equipotential circle at a right angle: $\ev{\vec{E} \perp \text{equipotential surfaces}}$ This follows from $\vec{E} = -\nabla V$: the field points in the direction of maximum variation of the potential, i.e. perpendicular to the surfaces $V = \text{const}$. Along an equipotential the field does no work.

(d) The dipole case. Replacing the sphere with a dipole $+Q/-Q$, the spherical symmetry is lost: the field lines leave $+Q$ and enter $-Q$ curving in arcs; the equipotentials are no longer concentric circles but deformed closed curves (ovals) wrapping around each charge, with a $V=0$ equipotential coinciding with the median plane perpendicular to the dipole segment. Even in this case, at every point, $\vec{E}$ remains orthogonal to the equipotential passing through it. Epstein’s graphical constructions show nicely how the “field lines / equipotentials” network forms an orthogonal grid everywhere.

See Riferimenti bibliografici for (Epstein 2002).

Links

Topics: Electric field and potential Concepts: Electric potential · Conductors in electrostatic equilibrium · Electric dipole Methods: Gradient method Objects: Charged sphere