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 Example $23.8 .$ Show that the electric field at distances $x$ that are large compared with $R$ approaches that of a particle with charge $Q=\sigma \pi R^{2} .$ Suggestion: 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 $\delta<<1$.

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Key Concepts

Electric Field of a Continuous Charge Distribution

This concept involves determining the electric field generated by a spatially extended source of charge by integrating small charge elements using Coulomb's law. The symmetry of the charged disk simplifies the integration along the axis, resulting in an expression that depends on both the distance and the disk’s radius.

Far-Field (Asymptotic) Approximation

When the observation point is located far from the charge distribution relative to its dimensions, the detailed structure of the distribution becomes negligible. In this limit, the field approximates that of a point charge, obeying an inverse square law that greatly simplifies the analysis and connection to fundamental electrostatic principles.

Binomial Expansion

The binomial expansion is a mathematical tool used to approximate expressions of the form (1 + ?)^n when ? is small. In the context of this problem, it helps simplify square-root expressions that arise in the electric field calculation, allowing for easier extraction of the dominant behavior at large distances.

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