dielectric of capacitors

Question

a dielectric with the dielectric constant k=4 is inserted into a parallel plate capacitor, filling 1/3 of the volume. if the capacitance of the capacitor without the dielectric is c whats is the capacitance with the dielectric?

Answer 1

General formula for a parallel plate capacitor:

C = kε₀A/d

C = capacitance (in Farads, F)

k = relative permittivity of dielectric material (unitless)

ε₀ = permittivity of free-space (=8.85x10-12 F/m)

A = area of plates (m²)

d = distance between the plates (m)

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Assuming the unmodified capacitor had an air gap (k=1 for air):

C = c = ε₀A/d (1)

Now, 1/3 of the volume is filled with a dielectric with k=4. Assuming this material exactly fills the gap, it will also fill 1/3 of the area. We can therefore find the new capacitance C_new as the sum of two capacitors: C_new = C₁ + C₂, where C₁ has an air-gap and C₂ has the k=4 material:

C_new = 1*ε₀*((2/3)A/d) + 4*ε₀*((1/3)A/d)

= (1/3)( 2ε₀(A/d) + 4ε₀(A/d))

// Making use of (1), re-write in terms of c:

= (1/3)( 2c + 4c )

= 2c

C_new = 2c (inserting the dielectric with k=4 to fill the 1/3^rd of the gap has the effect of doubling the capacitance)