Consider a parallel plate capacitor with rectangular plates and a sheet of metal

Question

Consider a parallel plate capacitor with rectangular plates and a sheet of metal of thickness a. The
dimensions of the capacitors are given in [mm]. Let x be the length of the metal plate that is
inserted between the capacitor plates. Let +Q and -Q be the charges [fC] on the plates of the capacitor.
a. Find the capacitance Co and the energy Uo stored in the capacitor before the metal sheet is inserted.
b. Find the capacitor C and the energy U stored in the capacitor after insertion of the metal sheet as function of x.
c. Find the force on the metal sheet? Which direction does it tend to move the metal sheet?

Explanation / Answer

a)capacitance C= Q/V =Ae/d (let e refer to ephsilon which is the permitivity) energy U= 0.5 CV^2 =0.5*e*A*V^2/b b)this set up effectively forms 3 capacitors two of them between plate of original capacitor and inserted plate and third one the original with reduced area 2 of them in series with capacitance of 0.5*e*A*x*V^2/l*d each effective capacitance of these two give C1= e*A*x*V^2/l*d third one in parallel two the effective capacitance of these two with capacitance C2= 0.5*e*A*(l-x)*V^2/l*d total capacitance C= C1*C2/(C1+C2) energy stored= 0.5 *C*v^2 c) net force on inserted plate is zero

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