Question 4. [15 marks] (a) Consider two parallel plates of area 2LLLLLLLLLLLLLLL

Question

Question 4. [15 marks] (a) Consider two parallel plates of area 2LLLLLLLLLLLLLLLLLLLL/////// +0 -0 A separated by air over a distance d. The plates are charged with charges +Q and -Q. The upper plate is fixed to a ceiling while a mass m is attached to the bottom plate. If the plates are in static equilibrium, find the mass m as a function of the other parameters (assume the weight of the lower plate is negligible). z=0 Hint: find the energy stored in the capacitor as a function of bottom plate position z and use the force-energy relationship =-du/dz to find the electrostatic force on the lower plate. (b) Assuming the mass is equal to the mass calculated in (a), what would happen if a dielectric of constant > 1 was inserted between the plates? Would the bottom plate remain in equilibrium or fall downwards? Justify your answer.

Explanation / Answer

a. consider two parallel plates

area = A

distacne between plates = d

charges = +Q, -Q

hence to find potential energy of the bottom plate as a function of z

PE = -mg(z + l) + 0.5*Q^2*z/epsilon*A

where l is length of the string attaching mass m to the plate

hence

F = -dPE/dz

F = mg - 0.5*Q^2/epsilon*A

for equilibrium, F = 0

hence

m = 0.5*Q^2/epsilon*A*g

b. if dielectric of k > 1 is inserted into the capacitor plates

the permittivity of free space to be used in the equation becomes relative permittivity = k*epsilon

hence the mass which can be balanced now decreases

hence the mass already hung by the cpaacitor plate falls down

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