Two very large horizontal sheets are 4.25 cm apart and carry equal but opposite uniform surface

Two very large horizontal sheets are 4.25 cm apart and carry...

Two very large horizontal sheets are 4.25 $\mathrm{cm}$ apart and carry equal but opposite uniform surface charge densities of magnitude $\sigma .$ You want to use these sheets to hold stationary in the region between them an oil droplet of mass 324$\mu$ that carries an excess of five electrons. Assuming that the drop is in vacuum, (a) which way should the electric field between the plates point, and (b) what should $\sigma$ be?

Two very large horizontal sheets are 4.25 $\mathrm{cm}$ apart and carry equal but opposite uniform surface charge densities of magnitude $\sigma .$ You want to use these sheets to hold stationary in the
region between them an oil droplet of mass 324$\mu$ that carries an excess of five electrons. Assuming that the drop is in vacuum, (a) which way should the electric field between the plates point, and (b) what should $\sigma$ be?

Key Concepts

Electric Field between Parallel Plates

This concept involves the understanding that two large, oppositely charged plates create a nearly uniform electric field in the space between them. The field is directed from the positive plate to the negative plate and its magnitude, in the ideal case of infinite plates, is given by the superposition of fields produced by each plate. This idea is central when analyzing the forces acting on any charge placed within the field.

Gauss’s Law

Gauss’s Law provides a relationship between the electric flux through a closed surface and the charge enclosed by that surface. It is crucial for deriving the electric field near charged surfaces, such as the sheets in a parallel plate setup, by relating the surface charge density to the resulting electric field.

Force on a Charged Particle

The fundamental relationship that a charged particle experiences a force when placed in an electric field is encapsulated by F = qE, where F is the force, q is the charge on the particle, and E is the electric field. This concept is important because it allows one to link the electric field created by the charged plates to the force necessary to balance other forces such as gravity.

Equilibrium of Forces

In many problems, achieving a stationary situation involves balancing forces acting on an object. In the context of charged particles in an electric field, equilibrium implies that the electric force acting on the particle must exactly counteract other forces such as gravitational force. This principle is used to determine the required field strength or surface charge density for equilibrium.

Surface Charge Density

Surface charge density, denoted by ?, describes the amount of electric charge per unit area on a surface. It is a key parameter in determining the magnitude of the electric field generated by a charged plate through its relationship with the permittivity of free space, and is integral to solving problems involving charged surfaces and fields.

Transcript

00:01 In this exercise, we have an oil droplet, this red particle in here, that lies between two electric plates of electric density sigma, and those electric plates are infinite and generate a uniform electric field.

00:21 What we want to know, what we want to calculate first, is what should be the direction of the uniform electric field, in order to the particle to be stationary between those two plates.

00:42 First, we must know that the particle is charged and its charge is negative.

00:47 It's five times the charge of the electron.

00:51 So first thing, we know that for this force to cancel the gravitational force, the force should be pointing upwards.

01:04 So we have that the electric field, the electric force, should be parallel to the positive wide direction.

01:18 But we know that the expression for the electric force is the charge of the particle times the electric field on which the particle is under.

01:34 We know that the charge is negative.

01:36 So in order to this quantity to be positive, the electric field must also be negative.

01:47 So the electric field must be pointing downwards to the negative wide direction.

01:55 So let's put the arrows here indicating that the electric field points downwards.

02:06 Okay.

02:07 Now question b asks us to calculate.

02:13 What should be the superficial density of charges on those planes in order to, so they can generate an electric field that maintains the droplet in equilibrium.

02:39 So first we can, well, so for the particle to be in equilibrium, in equilibrium, the forces must cancel, so the resultant force must be zero.

03:01 That implies that, since we only have forces in the wide direction, that the electric field force must be equal to the gravitational force, fg...

Please give Ace some feedback