Question

A 4.00 -pF parallel plate capacitor has a potential difference of \(10.0 \mathrm{~V}\) across it. The plates are \(3.00 \mathrm{~mm}\) apart, and the space between them contains air. a) What is the charge on the capacitor? b) How much energy is stored in the capacitor? c) What is the area of the plates? d) What would the capacitance of this capacitor be if the space between the plates were filled with polystyrene?

Short answer

Question: Calculate the charge, energy stored, area of the plates, and the capacitance when the space between the plates is filled with polystyrene for a parallel plate capacitor with capacitance 4 pF, potential difference 10 V, and distance between the plates 3.00 mm. Answer: The charge on the capacitor is \(4.00 \times 10^{-11} C\), the energy stored is \(2.00 \times 10^{-10} J\), the area of the plates is \(1.35 \times 10^{-3} m^2\), and the capacitance when the space between the plates is filled with polystyrene is \(10.3 pF\).

Step-by-step solution

a) Finding the charge on the capacitor

Since the capacitance C = 4 pF and the potential difference V = 10 V, we can calculate the charge using the formula: \(Q = CV\) \(Q = (4.00 \times 10^{-12} \mathrm{F})(10.0 \mathrm{V})\) \(Q = 4.00 \times 10^{-11} \mathrm{C}\)

b) Calculating the energy stored in the capacitor

With the known values of C and V, calculate the energy stored, using the formula: \(U = \frac{1}{2}CV^2\) \(U = \frac{1}{2}(4.00 \times 10^{-12} \mathrm{F})(10.0 \mathrm{V})^2\) \(U = 2.00 \times 10^{-10} \mathrm{J}\)

c) Finding the area of the plates

The formula for capacitance is: \(C = \frac{\epsilon_0 A}{d}\) To find the area A, rearrange the formula: \(A = \frac{Cd}{\epsilon_0}\) Where \(\epsilon_0 = 8.854 \times 10^{-12} \mathrm{F/m}\) and d = 3.00 mm (which we need to convert to meters) \(d = 3.00 \times 10^{-3} \mathrm{m}\) Now, calculate A as: \(A = \frac{(4.00 \times 10^{-12} \mathrm{F})(3.00 \times 10^{-3} \mathrm{m})}{(8.854 \times 10^{-12} \mathrm{F/m})}\) \(A = 1.35 \times 10^{-3} \mathrm{m^2}\)

d) Calculating capacitance with polystyrene

When the space between the plates is filled with polystyrene, the capacitance formula changes to: \(C' = \frac{\epsilon_r \epsilon_0 A}{d}\) Where \(\epsilon_r\) is the relative permittivity of polystyrene, approximately 2.56. Calculate the new capacitance as: \(C' = \frac{(2.56)(8.854 \times 10^{-12} \mathrm{F/m})(1.35 \times 10^{-3} \mathrm{m^2})}{(3.00 \times 10^{-3} \mathrm{m})}\) \(C' = 1.03 \times 10^{-11} \mathrm{F}\) or \(10.3 \mathrm{~pF}\)