A parallel plate capacitor has a capacitance C when there is no dieletric inside

Question

A parallel plate capacitor has a capacitance C when there is no dieletric inside of it. Suppose a wedge of material with dielectric constant K is inserted in between the plates of the capacitor (see figure). The bottom face of the wedge has the same area as the plate of the capacitor. The height of the wedge is equal to the thickness of the capacitor, t on the left edge and varies linearly until the height is zero on the right edge. What is the new capacitance with this dielectric inserted?

HINT: See if you can split up the capacitor into small capacitors that each have a dielectric in them that you know how to deal with.

A parallel plate capacitor has a capacitance C when there is no dieletric inside of it. Suppose a wedge of material with dielectric constant K is inserted in between the plates of the capacitor (see figure). The bottom face of the wedge has the same area as the plate of the capacitor. The height of the wedge is equal to the thickness of the capacitor, t on the left edge and varies linearly until the height is zero on the right edge. What is the new capacitance with this dielectric inserted? HINT: See if you can split up the capacitor into small capacitors that each have a dielectric in them that you know how to deal with

Explanation / Answer

Before the dielectric is inserted, the space between the plates is presumably filled with air. Since the dielectric constant of air is virtually indistinguishable from that of a vacuum, let us use the vacuum formula to calculate the initial capacitance Co. Thus,

Co=EoA/d

After the dielectric is inserted, the capacitance increases by a factor K, which in this case is the new capacitance C is given by

C=K.Co

Before the dielectric is inserted, the charge Qo on the plates is simply

Qo=Co.Vo

After the dielectric is inserted, the charge Q is exactly the same, since the capacitor is disconnected, and so the charge cannot leave the plates. Hence,

Qo=Q

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