Question

A 1.0-μm-diameter oil droplet (density 900 kg/m^3) is negatively charged with the addition of 27 extra electrons. It is released from rest 2.0 mm from a very wide plane of positive charge, after which it accelerates toward the plane and collides with a speed of 3.5 m/s . What is the surface charge density of the plane?

Answer 1

surface charge density = 5.91 µC/m²

Step-by-step explanation:

given data

diameter = 1 μm

radius = 0.5 ×
`10^(-6)` m

speed = 3.5 m/s

density 900 kg/m³

distance = 2 mm = 2 ×
`10^(-3)` m

to find out

surface charge density of the plane

solution

volume is express as

volume =
`(4)/(3) \pi r^3`

volume =
`(4)/(3) \pi (0.5*10^(-6))^3`

volume = 5.23 ×
`10^(-19)` m³

and

mass = density × volume

mass = 900 ×5.23×
`10^(-19)`

mass = 4.712 ×
`10^(-16)` kg

and

from motion of equation

v² -u² = 2×a×s

here v is speed 3.5 and u initial is 0 and a is acceleration and s is distance

3.5² = 2×a×2 ×
`10^(-3)`

acceleration = 3062.5 m/s²

so from newton second law

Force = mass × acceleration

force = 4.712 ×
`10^(-16)` ×3062.5

force = 1.44305 ×
`10^(-12)` N

so

surface charge density =
`(2\epsilon F)/(q)`

here q is charge with addition 27 extra electron and f is force and ∈ = 8.85×
`10^(-12)`

surface charge density =
`(2\epsilon F)/(q)`

surface charge density =
`(2*8.85*10^(-12)*1.443*10^(-12))/(27*1.60*10^(-19))`

surface charge density = 5.91 ×
`10^(-6)` C/m²

so surface charge density = 5.91 µC/m²