A 470-pF capacitor consists of two 15-cm-radius circular plates, insulated with polystyrene. Find (a) the thickness of the polystyrene and (b) the capacitor’s working voltage. Wolfson, Richard. Essential University Physics, Volume 2 (p. 447). Pearson Education. Kindle Edition.

(a) d = 3.46mm

(b) the capacitor’s working voltage =86.5 kV

Explanation:

a dielectric between capacitor plates increases the capacitance as determined from the equation  C = k, where k = dielectric constant

=

C = k

making d (spacing) as the subject of the formulae

d =k

the dielectric constant k  for polystyrene is 2.6

d = 2.6 × 8.85 pF/m × π (0.15)²/ 470pF

d = 1.62669195/470

d = 3.46 ×10∧ -3

the thickness of the polystyrene is therefore

d = 3.46mm

(b) the capacitor’s working voltage is find as

The dielectric breakdown for polystyrene is Emax=25kV/mm,

so the maximum voltage for this capacitor is Vmax=Emax ×d

=(25 kV/mm) (3.46 mm)=86.5 kV

(a) 0.0034m

(b) 81600V

The capacitance (C) of a capacitor with dielectric material is given by;

C = k x A x ε₀ / d   ------------------- (i)

Where;

k = dielectric constant of the dielectric material

ε₀ = permittivity of free space = 8.85 x 10⁻¹² F/m

A = area of one of the plates of the capacitor

d = distance between the plates (or thickness of the dielectric material)

From the question, the following are given;

radius (r) of the plates = 15cm = 0.15m

Capacitance (C) of the capacitor = 470pF = 470 x 10⁻¹² F

From the radius, lets calculate the area (A) of any of the plates as follows;

A = x

Take = 3.14, and substitute the value into the equation above.

A = 3.14 x 0.15²

A = 0.07065m²

Also, the dielectric constant (k) of polystyrene is about 2.56

(a) Now let's calculate the thickness of the dielectric (polystyrene).

Substitute the values of A, C, ε₀ and k into equation (i) as follows;

=> C = k x A x ε₀ / d

=> 470 x 10⁻¹² = 2.56 x 0.07065 x 8.85 x 10⁻¹² / d

Solve for d;

=> d =  2.56 x 0.07065 x 8.85 x 10⁻¹² / (470 x 10⁻¹²)

=> d = 0.0034m

Therefore the thickness of the dielectric material (polystyrene) is 0.0034m

(b) Now let's calculate the capacitor's working voltage.

The voltage (V) of the capacitor is related to the thickness (d) of its dielectric material as follows;

V = E x d   ----------------------------(ii)

Where;

d = 0.0034m

E = The dielectric strength of the dielectric material (polystyrene)

The dielectric strength of a material is the value of its electric field above which the material begins to conduct. Polystyrene's strength is 24 x 10⁶V/m

i.e E = 24 x 10⁶V/m

Substitute these values into equation (ii) as follows;

V = 24 x 10⁶ x 0.0034

V = 81600Volts

Therefore, the working voltage of the capacitor is 81600V

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