Which graph of E

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Which graph? Among the following trends of $|E|$ as a function of $r$ (distance from the axis of symmetry), which one correctly describes: (i) the field outside a uniformly charged sphere; (ii) the field between the plates of an ideal parallel-plate capacitor; (iii) the field generated by an infinite straight charged wire?

Answer by matching each situation to a letter and explain why in one line. Inspired by “Vie della Fisica” (Battimelli and Stilli 1999).

Solution

The three graphs show: A a $1/r^2$ trend, B a $1/r$ trend, C a constant. Applying Gauss’s theorem to the appropriate symmetry surface gives the dependence on $r$ for each case.

(i) Uniformly charged sphere (outside) → A. With a concentric spherical Gaussian surface of radius $r$, $E\cdot 4\pi r^2 = Q/\varepsilon_0$, hence $E = \frac{Q}{4\pi\varepsilon_0 r^2} \propto \frac{1}{r^2}$ This is graph A ($1/r^2$), the same as a point charge.

(ii) Ideal parallel-plate capacitor → C. Between the plates the field is uniform, $E = \sigma/\varepsilon_0$, independent of position: it does not depend on $r$. $E = \frac{\sigma}{\varepsilon_0} = \text{costante}$ This is graph C (constant).

(iii) Infinite straight wire → B. With a cylindrical Gaussian surface of radius $r$ and length $\ell$, $E\cdot 2\pi r\,\ell = \lambda\ell/\varepsilon_0$, hence $E = \frac{\lambda}{2\pi\varepsilon_0 r} \propto \frac{1}{r}$ This is graph B ($1/r$).

Final matching. $\ev{\text{(i)}\to\text{A}\ (1/r^2), \qquad \text{(ii)}\to\text{C}\ (\text{costante}), \qquad \text{(iii)}\to\text{B}\ (1/r)}$

This difference reflects the dimensionality of the symmetry: point/spherical ($1/r^2$), cylindrical ($1/r$), planar (constant). See Riferimenti bibliografici for (Battimelli and Stilli 1999).

Links

Topics: Electric field and potential Concepts: Electric field · Gauss’s theorem Skills: Application of Gauss’s theorem Objects: Charged sphere · Infinite straight wire · Parallel-plate capacitor