Problem — Reading a graph: V(r) of a point charge

Problem

Reading a graph: $V(r)$ of a point charge. A graph shows the potential $V(r)$ generated by a charge $+q$ as a function of the distance $r$. Describe qualitatively the shape of the curve and explain what an equipotential line represents geometrically for a point charge.

Solution

Shape of the curve. The potential of a positive point charge is

$V(r) = k\,\frac{q}{r} \ \propto\ \frac{1}{r}$

It is a hyperbola: $V \to +\infty$ as $r \to 0$ (very close to the charge the potential diverges) and $V \to 0$ as $r \to \infty$ (far away it vanishes). The curve decreases rapidly at first and then flattens out, approaching the horizontal axis as an asymptote.

Equipotential lines. An equipotential is the locus of points with the same $V$; since $V$ depends only on $r$, this locus is the set of points equidistant from the charge, i.e. a sphere centred on the charge (a circle in a planar section). The equipotentials are therefore concentric spheres, always perpendicular to the radial field lines.

$\ev{V \propto 1/r \text{ (iperbole)};\ \text{equipotenziali} = \text{sfere concentriche sulla carica}}$

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Topics: Campo elettrico e potenziale Concepts: Potenziale elettrico Skills: Lettura dei grafici