Question

A 1.0-μm-diameter oil droplet (density 900 kg/m3) is negatively charged with the addition of 39 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 4.5 m/s.

What is the surface charge density of the plane?

Answer 1

`6.75\mu C/m^2`

Step-by-step explanation:

We are given that

Diameter,d=
`1\mu m=1\time 10^(-6) m`

`1\mu m=10^(-6) m`

Radius,r=
`(d)/(2)=(1)/(2)* 10^(-6)=0.5 * 10^(-6) m`

Density,
`\rho=900kg/m^3`

Total number of electrons,n=39

Charge on electron =
`1.6* 10^(-19) C`

Total charge=
`q=ne=39* 1.6* 10^(-19)=62.4* 10^(-19) C`

Distance,s=2mm=
`2* 10^(-3) m`

Mass =
`density* volume=900* (4)/(3)\pi r^3=900* (4)/(3)\pi(0.5* 10^(-6))^3=4.7* 10^(-16) kg`

Initial velocity,u=0

Final speed,v=4.5 m/s

`v^2-u^2=2as`

`(4.5)^2-0=2a(2* 10^(-3))`

`20.25=4a* 10^(-3)`

`a=(20.25)/(4* 10^(-3))=5062.5m/s^2`

Force,F=ma

`qE=ma`

`q((\sigma)/(2\epsilon_0))=ma`

`\sigma=(2\epsilon_0ma)/(q)=(2* 8.85* 10^(-12)* 4.7* 10^(-16)* 5062.5)/(62.4* 10^(-19))`

`\epsilon_0=8.85* 10^(-12)`

`\sigma=6.75* 10^(-6)C/m^2=6.75\mu C/m^2`