When a shower is turned on in a closed bathroom, the...

When a shower is turned on in a closed bathroom, the splashing of the water on the bare tub can fill the room's air with negatively charged ions and produce an electric field in the air as great as 1000 $\mathrm{N} / \mathrm{C}$ . Consider a bathroom with dimensions 2.5 $\mathrm{m} \times$ 3.0 $\mathrm{m} \times 2.0 \mathrm{m}$ . Along the ceiling, floor, and four walls, approximate the electric field in the air as being directed perpendicular to the sur- face and as having a uniform magnitude of 600 $\mathrm{N} / \mathrm{C}$ Also, treat those surfaces as forming a closed Gaussian surface around the room's air. What are (a) the volume charge density $\rho$ and (b) the number of excess elementary charges e per cubic meter in the room's air?

When a shower is turned on in a closed bathroom, the
splashing of the water on the bare tub can fill the room's air with
negatively charged ions and produce an electric field in the air as
great as 1000 $\mathrm{N} / \mathrm{C}$ . Consider a bathroom with dimensions 2.5 $\mathrm{m} \times$
3.0 $\mathrm{m} \times 2.0 \mathrm{m}$ . Along the ceiling, floor, and four walls, approximate
the electric field in the air as being directed perpendicular to the sur-
face and as having a uniform magnitude of 600 $\mathrm{N} / \mathrm{C}$ Also, treat those
surfaces as forming a closed Gaussian surface around the room's air.
What are (a) the volume charge density $\rho$ and (b) the number of
excess elementary charges e per cubic meter in the room's air?

Key Concepts

Elementary Charge

The elementary charge is the basic unit of charge in physics, representing the charge carried by a single proton or the magnitude of the negative charge carried by an electron. It serves as the fundamental building block for all quantized electric charge, and when calculating the number of excess charges in a volume, the total charge is divided by the elementary charge to provide the number of discrete charged entities present.

Volume Charge Density

Volume charge density quantifies the amount of electric charge per unit volume within a region of space. It is defined as the total charge divided by the volume over which it is spread. This concept is important in characterizing continuous charge distributions and is used to relate the observed electric field to the underlying charge distribution within the material or space.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the net electric flux passing through a closed surface to the charge enclosed within that surface. It is mathematically expressed as the integral of the electric field over a closed surface equaling the enclosed charge divided by the permittivity of free space. This concept is commonly used to simplify the calculation of electric fields when there is high symmetry in the charge distribution.

Gaussian Surface

A Gaussian surface is an imaginary closed surface used in conjunction with Gauss's Law to evaluate the electric field and the charge distribution. The selection of an appropriate Gaussian surface leverages the symmetry of the problem, allowing the electric field to be treated as uniform over parts of the surface, thereby simplifying the flux calculation.

Electric Field

The electric field represents the force per unit charge at a given point in space due to electric charges. It is a vector field that determines the direction and magnitude of the force acting on a test charge placed within the field. Understanding its distribution is crucial for predicting the behavior of charged particles and analyzing electrical interactions.

Transcript

00:01 Problem 23 .8, we're told that if you turn a shower on in a closed bathroom, the splashing of the water on the sides of the bathtub or shower stall, can fill the rooms there with negatively charged ions, which produce, of course, an electric field.

00:20 So let's imagine we have a bathroom that has the dimensions of 2 .5 meters times three meters by another 2 meters.

00:37 We have an electric field of 600 newtons per coolome along, it's constant along each of the walls and floor and ceiling.

00:56 And it's perpendicular to that, so this is the right thing.

01:01 So we wanna know what is the volume charge density and basically the number of excess elementary charges per unit volume.

01:24 So the total surface area, founding the bathroom we you know just you have two surfaces each with dimensions of these you square each of these add them up multiply that by two because they're two of each surface and we get that we have a 37 meters squared surface under consideration so then multiplying this by our 600 newtons per cullome we get 22 times 10 to the third newt meters squared per cullome now the the sign of the enclosed chart and this is just the absolute value of the flux because we're not told what which direction inward or outward the 600 newtons per coulome is directed, we're just told it's perpendicular to the walls and ceiling and floor.

02:54 So then this is going to equal epsilon not, which is hopefully always positive, since it's supposed to be constant, times the flux.

03:08 So this will be two times ten to the negative seventh coulomes.

03:18 The volume, just multiply these together now, is 15 cubic meters.

03:30 Take the charge divided by the volume...

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