Chapter 23, Problem 008 Flying Circus of Physics When a shower is turned on in a

Question

Chapter 23, Problem 008 Flying Circus of Physics 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 N/C. Consider a bathroom with dimensions 2.7 m x 4.1 m x 1.5 m. Along the ceiling, floor, and four walls, approximate the electric field in the air as being directed perpendicular to the surface and as having a uniform magnitude of 675 N/C. Also, treat those surfaces as forming a closed Gaussian surface around the room's air. What are (a) the volume charge density p and (b) the number of excess elementary charges e per cubic meter in the room's air? (a) Number Units (b) Number Units SHOW HINT

Explanation / Answer

The surface area of the bathroom will be:

A = 2 x 2.7 x 4.1 + 2 x 4.1 x 1.5 + 2 x 1.5 x 2.7 = 42.54 m^2

We know that the flux is:

phi = E A = 1000 x 42.54 = 4.25 x 10^4 N m^2/C

Also, from Gauss law, phi = q/e0

q = phi e0

q = 4.25 x 10^4 x 8.85 x 10^-12 = 3.76 x 10^-7 C

volume of the room is:

V = 2.7 x 4.1 x 1.5 = 16.61 m^3

rho = -3.76 x 10^-7/16.61 = -2.26 x 10^-8 C/m^3

Hence, rho = -2.26 x 10^-8 C/m^3

b)excess elemenatary charges can be calculated as follows:

n = 3.76 x 10^-7/(1.6 x 10^-19)/16.61 = 2.35 x 10^12 = 1.41 x 10^11

Hence, n = 1.41 x 10^11

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