A thin sheet of charge is in the x-y plane at z=0. It is...

A thin sheet of charge is in the $x-y$ plane at $z=0$. It is infinite but not uniformly charged. A circular piece of radius $R$ with its centre at the origin has a surface charge density $+\sigma$ while the rest of the sheet carries a surface charge density $-\sigma$. Show that the electric field vanishes at points located at $(0,0,+\sqrt{3} R)$ and $(0,0,-\sqrt{3} R)$. [Hint: The sheet can be regarded as an infinite sheet of uniform surface charge density plus a circular sheet with a different uniform surface charge density]

Key Concepts

Superposition Principle

The superposition principle states that the resultant electric field created by a distribution of charges is the vector sum of the fields produced by each charge or charge element individually. In this problem, the non?uniform sheet is decomposed into an infinite uniformly charged sheet and a circular patch with an opposing charge density, allowing for the individual and easier calculation of fields that can then be added together.

Electric Field of an Infinite Charged Sheet

An infinite sheet with a uniform surface charge density produces an electric field of constant magnitude at any point in space (neglecting edge effects), given by ?/(2??), with the field directed normally away from the sheet. This concept is fundamental for analyzing the contribution from the vast part of the sheet that is uniformly charged.

Electric Field on the Axis of a Charged Disk

The electric field due to a uniformly charged circular disk along its central axis can be derived through integration or by using a well-known formula. It varies with both the radius of the disk and the distance from the disk, and understanding this behavior is essential for evaluating the effect of the finite circular patch with a different charge density.

Symmetry in Electrostatics

Symmetry in an electrostatic problem simplifies the analysis of electric fields. In systems with cylindrical or planar symmetry, the electric field components from charge distributions can cancel or reinforce each other in predictable ways. In this scenario, the symmetry about the z-axis is crucial in demonstrating that the fields cancel out at certain points, leading to a zero net electric field.

Transcript

Please give Ace some feedback