A point charge of mass 0.210 kg, and net charge +0.340 μC,...

A point charge of mass $0.210 \mathrm{~kg}$, and net charge $+0.340 \mu \mathrm{C}$, hangs at rest at the end of an insulating cord above a large sheet of charge. The horizontal sheet of fixed uniform charge creates a uniform vertical electric field in the vicinity of the point charge. The tension in the cord is measured to be $5.18 \mathrm{~N}$. (a) Calculate the magnitude and direction of the electric field due to the sheet of charge (Fig. $21-79) .(b)$ What is the surface charge density $\sigma\left(\mathrm{C} / \mathrm{m}^{2}\right)$ on the sheet?

Key Concepts

Force Equilibrium in Electrostatics

Force equilibrium is achieved when the net force acting on an object is zero, implying that the object is either at rest or moving with constant velocity. In the context of this problem, the point charge hangs at rest, meaning that all the forces (electric force, gravitational force, and the tension in the cord) are balanced. This principle allows us to set up equations that relate the different forces, thereby enabling the calculation of unknown quantities like the magnitude of the electric field or the surface charge density.

Surface Charge Density

Surface charge density, usually denoted by sigma (?), represents the amount of charge per unit area on a charged surface. It directly influences the magnitude of the electric field produced by the charged sheet. When the field is produced by an infinite or sufficiently large sheet of charge, the relation E = ?/(2?0) can be used, where ?0 is the permittivity of free space. This concept is fundamental in linking the physical distribution of charge with the resulting electric field.

Electrostatic Force

This concept involves the force experienced by a charged particle when placed in an electric field. The force is given by F = qE, where q is the charge of the particle and E is the electric field strength. In problems like this, it is essential to understand that the net force on the charge results from the vector sum of the gravitational force, the tension in the cord, and the electric force acting on the charged particle.

Uniform Electric Field

A uniform electric field is one in which the field strength and direction are constant at every point in the region. In this problem, the large charged sheet creates a uniform electric field in its vicinity. Understanding uniform fields is key in electrostatics, as they simplify the analysis of forces and potential differences, allowing for direct application of the relationship F = qE without having to integrate over positions.

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