Question 25.43: Given a 7.4 pF air-filled capacitor, you are asked to conver......

Fundamentals of Physics Extended – Instructor’s Solutions Manual [1360823]

Given a 7.4 pF air-filled capacitor, you are asked to convert it to a capacitor that can store up to 7.4 μJ with a maximum potential difference of 652 V. Which dielectric in Table 25-1 should you use to fill the gap in the capacitor if you do not allow for a margin of error?

Table 25-1 Some Properties of Dielectrics $^a$
Material	Dielectric Constant $\kappa$	Dielectric Strength (kV/mm)
Air (1 atm)	1.00054	3
Polystyrene	2.6	24
Paper	3.5	16
Transformer oil	4.5
Pyrex	4.7	14
Ruby mica	5.4
Porcelain	6.5
Silicon	12
Germanium	16
Ethanol	25
Water (20°C)	80.4
Water (25°C)	78.5
Titania ceramic	130
Strontium titanate	310	8
For a vacuum, $\kappa$ = unity.
$^a$ Measured at room temperature, except for the water.

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The capacitance with the dielectric in place is given by $C=\kappa C_0$ , where C $_0$ is the capacitance before the dielectric is inserted. The energy stored is given by $U=\frac{1}{2} C V^2=\frac{1}{2} \kappa C_0 V^2$ , so

$\kappa=\frac{2 U}{C_0 V^2}=\frac{2\left(7.4 \times 10^{-6} \,J \right)}{\left(7.4 \times 10^{-12} \,F \right)(652 \,V )^2}=4.7$ .

According to Table 25-1, you should use Pyrex.

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