Answer:
(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[tex]C{o}[/tex], where k = dielectric constant
[tex]C{o}[/tex] = [tex]\frac{EoA}{d}[/tex]
C = k[tex]\frac{EoA}{d}[/tex]
making d (spacing) as the subject of the formulae
d =k[tex]\frac{EoA}{C}[/tex]
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 = [tex]\pi[/tex] x [tex]r^{2}[/tex]
Take [tex]\pi[/tex] = 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
