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Chapter 18: Problem 55

A charge of \(2.0 \times 10^{-10} \mathrm{C}\) is to be stored on each plate of a parallel-plate capacitor having an area of \(650 \mathrm{mm}^{2}\left(1.0 \mathrm{in.}^{2}\right)\) and a plate separation of \(4.0 \mathrm{mm}(0.16 \mathrm{in.})\) (a) What voltage is required if a material having a dielectric constant of 3.5 is positioned within the plates?

Short Answer

Expert verified
Answer: The required voltage is approximately \(38.8 \mathrm{V}\).

Step by step solution

01

Identify the given values

We are given the following values: - Charge on each plate (\(Q\)): \(2.0 \times 10^{-10} \mathrm{C}\) - Area of each plate (\(A\)): \(650 \mathrm{mm}^{2} = 650 \times 10^{-6} \mathrm{m}^{2}\) - Plate separation (\(d\)): \(4.0 \mathrm{mm} = 4.0 \times 10^{-3} \mathrm{m}\) - Dielectric constant (\(\epsilon_r\)): \(3.5\)
02

Calculate the capacitance

The capacitance (\(C\)) of the parallel-plate capacitor can be calculated using the formula: \(C = \frac{\epsilon_r \epsilon_0 A}{d}\) Here, \(\epsilon_0\) is the vacuum permittivity, equal to \(8.85 \times 10^{-12} \mathrm{F/m}\). Now plug in the given values: \(C = \frac{3.5 \times 8.85 \times 10^{-12} \mathrm{F/m} \times 650 \times 10^{-6} \mathrm{m}^{2}}{4.0 \times 10^{-3} \mathrm{m}}\) \(C = 5.15 \times 10^{-12} \mathrm{F}\)
03

Calculate the required voltage

We can find the required voltage (\(V\)) using the relationship between charge, capacitance, and voltage: \(V = \frac{Q}{C}\) Plug in the given charge and calculated capacitance: \(V = \frac{2.0 \times 10^{-10} \mathrm{C}}{5.15 \times 10^{-12} \mathrm{F}}\) \(V \approx 38.8 \mathrm{V}\)
04

Conclusion

To store a charge of \(2.0 \times 10^{-10} \mathrm{C}\) on each plate of the parallel-plate capacitor with an area of \(650 \mathrm{mm}^{2}\), a plate separation of \(4.0 \mathrm{mm}\), and a dielectric constant of \(3.5\), a voltage of approximately \(38.8 \mathrm{V}\) is required.

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Most popular questions from this chapter

For each of the following pairs of semiconductors, decide which will have the smaller band gap energy, \(E_{g},\) and then cite the reason for your choice. (a) \(C\) (diamond) and Ge, (b) AlP and InSb, (c) GaAs and ZnSe, (d) ZnSe and CdTe, and (e) \(\mathrm{CdS}\) and \(\mathrm{NaCl}\)

For \(\mathrm{CaO}\), the ionic radii for \(\mathrm{Ca}^{2+}\) and \(\mathrm{O}^{2}\) ions are 0.100 and \(0.140 \mathrm{nm},\) respectively. If an externally applied electric field produces a \(5 \%\) expansion of the lattice, compute the dipole moment for each \(\mathrm{Ca}^{2+}-\mathrm{O}^{2-}\) pair. Assume that this material is completely unpolarized in the abscnce of an electric field.

Is it possible for compound semiconductors to exhibit intrinsic behavior? Explain your answer.

The polarization \(P\) of a dielectric material positioned within a parallel- plate capacitor is to be \(4.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2}\) (a) What must be the dielectric constant if an electric field of \(10^{5} \mathrm{V} / \mathrm{m}\) is applied? (b) What will be the dielectric displacement \(D ?\)

Briefly describe electron and hole motions in a \(p-n\) junction for forward and reverse biases; then explain how these lead to rectification.
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