Physics · Ch 2 — Electrostatic Potential and Capacitance

Equipotential Surfaces and Their Properties

2.7

Equipotential Surfaces and Their Properties

An equipotential surface is the set of all points in space that share exactly the same value of electric potential. Rather than describing the field point by point with individual field-vector arrows, an equipotential surface joins up an entire continuous surface's worth of points at once, giving a geometric picture of how potential (and, indirectly, field) varies through space.

Property 1: no work is done moving a charge along an equipotential surface. Since every point on the surface shares the same value of $V$ , the potential difference between any two points ON the surface is exactly zero, and so the work done, $W = q\,\Delta V$ , in moving a charge between them is also exactly zero -- regardless of the path taken across the surface.

Property 2: the electric field is always perpendicular to an equipotential surface, at every point. If the field had any component tangential to the surface, that tangential component would do nonzero work moving a charge along the surface, directly contradicting Property 1; the only way to guarantee zero work along the entire surface is for the field to have no tangential component anywhere on it, i.e. to point exactly perpendicular (normal) to the surface at every point.

Property 3: no two equipotential surfaces can ever intersect. A single point in space has one, single, well-defined value of potential -- it cannot simultaneously belong to two DIFFERENT equipotential surfaces (which, by definition, correspond to two different values of $V$ ). If two equipotential surfaces did cross at a point, that point would need two different field directions (each perpendicular to its own surface, by Property 2) at once, which is impossible for a single-valued field.

Property 4: equipotential surfaces are spaced closer together where the field is strong, and farther apart where the field is weak. From $E = -dV/dl$ (Section 2.3), a LARGER field magnitude $E$ requires a SMALLER spatial step $dl$ to produce the same fixed change $dV$ between one equipotential surface and the next -- so tightly bunched surfaces mark a strong-field region, and widely spaced surfaces mark a weak-field region, exactly as the accompanying figures show for a point charge and for a dipole. …

Figure 1Equipotential surfaces around an isolated point charge

What this figure shows. A single positive point charge $q$ is drawn at the centre of the figure. Around it, several concentric dashed circles (representing, in three dimensions, concentric spheres) are drawn at increasing radii from the charge, each circle labelled with a potential value that decreases steadily from the innermost circle outward -- for instance the innermost drawn circle might be labelled $V_1$ , the next $V_2$ , and so on with $V_1 > V_2 > V_3$ , since potential due to a positive point charge falls off with distance. The circles are spaced perceptibly CLOSER together near the charge and progressively FARTHER apart at larger radii, showing that equal steps in potential correspond to unequal steps in radius -- closely spaced surfaces mark a region of strong field, widely spaced surfaces a region of weak field. A short radial arrow is drawn from the charge outward, crossing every circle at right angles, labelled $\vec{E}$ , to emphas …

Figure 2Equipotential surfaces of an electric dipole

What this figure shows. A dipole is drawn as a positive charge $+q$ on the left and an equal negative charge $-q$ on the right, separated by a small gap, with the dipole moment vector $\vec{p}$ drawn as a short arrow pointing from $-q$ to $+q$ . Around this pair, a family of closed, non-circular curves is sketched (representing, in three dimensions, closed but non-spherical equipotential surfaces): the curves are noticeably egg-shaped and bunched tightly around each individual charge, close to it, where the surfaces are nearly circular (looking locally like the equipotentials of a single point charge, since one charge dominates there); farther from the pair, in the mid-region between and around the two charges, the curves elongate and distort into a more complex, dumbbell-like shape rather than resembling either single-charge pattern; and directly on the perpendicular bisector plane of the dipole (equidistant from $+q$ and $-q$ ), an equipotential surface for $V = 0$ is drawn as a flat plane perpendicular to $\vec{p}$ passing th …

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