Physics Gaussian Law

Question

A thin conductive sheet is placed on the ground. The sheet is square with L m side
length and carries a total charge of Q C. A small ball bearing is held above the centre of the
sheet and carries a charge of q C. The ball bearing can be considered as a point charge with a
mass of m kg. The sheet can be considered as infinite in extent compared with the bearing.

 

1) Derive the mass of the ball bearing that will allow it to levitate above the
conductive sheet.

 

2) If the bearing is held at the surface and released, calculate its initial acceleration in
the vertical direction. Neglect both gravity and air resistance in this case. 

 

Q =25 C q = 1 x 10^-3 C  L= 100 m   and m =10g

 

 

For part 1 I am sure its equating 2 forces and solving for m? Not 100% sure.

 

No idea how to do part 2)

 

Thanks

 

Ian

Answer 1

(1)

 

If the sheet is infinite in extent, as stated near the end of the problem, its electric field is vertical and its magnitude is constant.  You should have seen the derivation of this in class.

 

E = σ / (2ε₀)

 

Note E is uniform everywhere.

 

σ =  surface charge density = Q/A = Q/L²

 

E = Q / (2ε₀L²)

 

F = electric force

 

F = qE

 

If the point charge is levitated, F must balance the weight.

 

qE = mg

 

m = qE / g = (q / g) [Q / (2ε₀L²)]

 

Plug in the numbers and get m. 

 

(2)

 

Use Newton's 2nd law:

 

a = F_t/m

 

F_t = net vertical force

 

Here use the mass given in part (2), which may be different from the mass you calculated in part (1).  E is the same, since E depends only on Q and L.

 

F_t = qE - mg

 

a = (qE - mg) / m

 

If you ignore gravity, then

 

a = qE / m

 

Can you finish from here?