When a shower is turned on in a dosed bathroom the splashing

When a shower is turned on in a dosed 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 1000.0 N/C. Consider a bathroom with dimensions 2.5 m Times 3.0 m Times 2.0 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 750.0 N/C. Also, treat those surfaces as forming a closed Gaussian surface around the room\'s air. [Place your diagram above this text.] What is the volume charge density rho?

Solution

The total surface area bounding the bathroom is

A= 2(2.5*3)+2(3*2)+2(2*2.5) = 37 m²

The absolute value of the total electric flux, with the assumptions stated in the problem, is

= E.A = 750*37 = 27750 Nm²/C

By Gauss’ law, the enclosed charge (in absolute value) is

     q = ₀ = 2.45 x 10^-7 C

volume V = 15 m³

the charge density is q_enc/V = –1.64 ´ 10^–8 C/m³.