The electric field along the axis of a uniformly charged...

The electric field along the axis of a uniformly charged disk of radius $R$ and total charge $Q$ was calculated in $\mathrm{Ex}-$ ample 23.9. Show that the electric field at distances $x$ that are great compared with $R$ approaches that of a point charge $Q=\sigma \pi R^{2}$. (Hint: First show that $x /\left(x^{2}+R^{2}\right)^{1 / 2}=\left(1+R^{2} / x^{2}\right)^{-1 / 2}$, and use the binomial expansion $(1+\delta)^{n} \approx 1+n \delta$ when $\left.\delta \ll 1 .\right)$

Key Concepts

Uniform Charge Distribution

This concept involves understanding that the charge is spread evenly over a surface area, which is fundamental in calculating the resultant electric field. Rather than having discrete point charges, the continuous distribution allows us to use integration techniques to sum the contributions of infinitesimal charge elements across the disk, taking advantage of the system’s symmetry to simplify the problem.

Electric Field of a Charged Disk

The calculation of the electric field along the axis of a uniformly charged disk is a standard problem in electromagnetism. It requires integrating the contributions from all parts of the disk, considering the geometry of the problem. The resulting expression typically involves the disk's radius R, the distance x along the axis, and the surface charge density, leading to an electric field that depends on these parameters in a nontrivial way.

Far-Field Approximation

In many physical situations, it is useful to examine the behavior of the electric field at distances much larger than the size of the charge distribution (x >> R). Under these conditions, the charged disk can be approximated by a point charge. This approximation simplifies the mathematical treatment by reducing a complex, spatially extended charge distribution to a simpler point-like source, which is easier to analyze.

Binomial Expansion

The binomial expansion is a mathematical tool used to approximate expressions of the form (1 + ?)^n when ? is much smaller than 1. In the context of the problem, it allows the simplification of the electric field formula for the charged disk when the term R²/x² is small. This approximation shows that the correction to the point charge field is negligible in the far-field limit, reinforcing the idea that the disk's field approaches that of a point charge Q.

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