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Chapter 24: Problem 29

Calculate the capacitance of the Earth. Treat the Earth as an isolated spherical conductor of radius \(6371 \mathrm{~km}\).

Short Answer

Expert verified
Answer: The approximate capacitance of the Earth is 715.4 pF (picoFarads).

Step by step solution

01

Convert radius to meters

First, we need to convert the given radius from kilometers to meters: $$ 6371 \mathrm{~km} \times \frac{1000 \mathrm{~m}}{1 \mathrm{~km}} = 6,371,000 \mathrm{~m} $$ The radius of the Earth in meters is now \(6,371,000 \mathrm{~m}\).
02

Calculate capacitance

Now, we can plug the values into the capacitance formula: $$ C = 4\pi \epsilon_0 R $$ Where \(\epsilon_0 = 8.854\times 10^{-12} \mathrm{F/m}\) is the vacuum permittivity. Substituting the values, we get: $$ C = 4\pi (8.854\times 10^{-12} \mathrm{F/m}) (6,371,000 \mathrm{~m}) $$
03

Compute the result

Now, we can perform the calculations to find the capacitance: $$ C \approx 4 \times 3.14 \times (8.854\times 10^{-12} \mathrm{F/m}) \times (6,371,000 \mathrm{~m}) $$ $$ C \approx 715.4 \times 10^{-12} \mathrm{F} $$ Thus, the capacitance of the Earth can be approximated as \(715.4 \times 10^{-12} \mathrm{F}\) or \(715.4 \mathrm{pF}\) (picoFarads).

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Most popular questions from this chapter

Two capacitors, with capacitances \(C_{1}\) and \(C_{2},\) are connected in series. A potential difference, \(V_{0}\), is applied across the combination of capacitors. Find the potential differences \(V_{1}\) and \(V_{2}\) across the individual capacitors, in terms of \(V_{0}\), \(C_{1},\) and \(C_{2}\).

A parallel plate capacitor has a capacitance of \(120 .\) pF and a plate area of \(100 . \mathrm{cm}^{2}\). The space between the plates is filled with mica whose dielectric constant is \(5.40 .\) The plates of the capacitor are kept at \(50.0 \mathrm{~V}\) a) What is the strength of the electric field in the mica? b) What is the amount of free charge on the plates? c) What is the amount of charge induced on the mica?

Thermocoax is a type of coaxial cable used for high-frequency filtering in cryogenic quantum computing experiments. Its stainless steel shield has an inner diameter of \(0.35 \mathrm{~mm},\) and its Nichrome conductor has a diameter of \(0.17 \mathrm{~mm}\). Nichrome is used because its resistance doesn't change much in going from room temperature to near absolute zero. The insulating dielectric is magnesium oxide \((\mathrm{MgO}),\) which has a dielectric constant of \(9.7 .\) Calculate the capacitance per meter of Thermocoax.

Two capacitors with capacitances \(C_{1}\) and \(C_{2}\) are connected in series. Show that, no matter what the values of \(C_{1}\) and \(C_{2}\) are, the equivalent capacitance is always less than the smaller of the two capacitances.

A capacitor with a vacuum between its plates is connected to a battery and then the gap is filled with Mylar. By what percentage is its energy-storing capacity increased?
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