A capacitor is constructed from two square metallic plates of sides e and separa
ID: 2037885 • Letter: A
Question
A capacitor is constructed from two square metallic plates of sides e and separation d Charges +Q and Qare piace on the plates and the power supply ist en removed material of dielectric constant ? is inserted a distance x into the capacitor as shown in the figure below. Assume d is m, uch smaller than x. Suggestion: The system can be considered as two capacitors connected in parallel. (Use the following as necessary: ?0, K, e, d, and x.) +Q (a) Find the equivalent capacitance of the device Ceq (b) Calculate the energy stored in the capacitor (c) Find the direction and magnitude of the force exerted by the plates on the dielectric magnitude direction | Direction# (d) Obtain a numerical value for the force when x - £/2, assuming -4.90 cm, d-1.90 mm, the dielectric is glass (4.50), and the capacitor was charged to 1.90 x 103 V before the dielectric was inserted. magnitude directionDirection UNExplanation / Answer
pplied. We can calculate the capacitance of each, and the total equivalent,
capacitance easily:
a)
Cfilled = K*E0 * Afilled / d = K*E0*l*x / d
Cempty = E0*Aempty / d = E0 (l - x) *l / d
C eq = Cfilled + Cempty
= E0*l / d [(k-1)x + l]
= E0*l^2/d [x/l (k-1) +1]
b)
The last form is perhaps more pleasing, since it tells us the equivalent capacitance compared to having
no dielectric at all E0*l^2/d. The total energy stored can now be found easily from the equivalent
capacitance and voltage. Remember: an equivalent capacitance is equivalent in every way, so the
energy in the equivalent capacitor is the same as that in the individual constituents.
U = 1/2 Ceq (?V)^2
= 1/2 [ E0 * l^2 ((?V)^2 / d ] [x/l (k-1) +1]
c)
F = - dU/dx
= - E0*l / 2d [ k -1 ] (?V)^2
The force in this case acts to the right, pulling the sheet in. This is because the dielectric is polarizable
- the top surface of the dielectric near the positive plate would develop a negative charge, and be
attracted to the capacitor, pulling the dielectric farther in. Another way to think about it is that the
capacitor stores more energy with the dielectric inside, so it will try to pull it in and maximize its
stored energy
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