Two ways to calculate the field

From the preceding concepts — field lines, line density and equipotentials — two complementary strategies emerge for finding the field $\vec{E}$ at a point. The first starts from the sources and the field lines, the second from the potential and the equipotentials: they are two doors into the same problem, and the choice depends on which information we have available.

Way 1 — Perpendicular surface (field lines)

This is the direct method of Faraday’s paradigm:

1. Draw (or imagine) the field lines in the region of interest.
2. Choose a small surface $S_\perp$ perpendicular to the lines at the desired point.
3. Count $N$, the equivalent charge in coulombs of the lines crossing it.
4. Calculate $|\vec{E}| = 4\pi k\, N/S_\perp$.

When the problem has a symmetry — spherical, cylindrical or planar — $S_\perp$ can be chosen so that the entire flux reduces to the product $|\vec{E}| \cdot S_\perp$: this is exactly what Gauss’s theorem does. Symmetry does the heavy lifting, allowing $|\vec{E}|$ to be taken out of the sum over the line contributions.

Way 2 — Gradient of the potential

If $V$ is known throughout space (or the equipotentials can be drawn), we use $|\vec{E}| \approx \frac{|\Delta V|}{\Delta d}$ where $\Delta d$ is the distance between two nearby equipotentials with difference $|\Delta V|$. The direction of $\vec{E}$ is perpendicular to the equipotential, in the direction in which $V$ decreases.

Way 2 is often the most convenient when a potential map is available: the field strength is read from the spacing of the equipotentials (denser, stronger field) and the direction from their perpendicular. It is the link between field and potential read in reverse with respect to the definition of $V$.

Warning

$\vec{E}$ always points from high potential towards low potential — like water flowing downhill. In formal terms, $\vec{E}$ is the “downhill gradient” of $V$, with the minus sign: the field indicates the direction along which the potential decreases most rapidly.

Collegamenti

Argomenti: Campo elettrico e potenziale · Elettrostatica Concetti: Campo elettrico · Potenziale elettrico

Esercizi collegati: True or false on field and potential · Cloud-to-ground potential difference · Field lines never cross