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Chapter 24: Problem 30

Two concentric metal spheres are found to have a potential difference of \(900 . \mathrm{V}\) when a charge of \(6.726 \cdot 10^{-8} \mathrm{C}\) is applied to them. The radius of the outer sphere is \(0.210 \mathrm{~m}\). What is the radius of the inner sphere?

Short Answer

Expert verified
Answer: The radius of the inner sphere is approximately 0.141 m.

Step by step solution

01

Understand the given information and what we need to find

We are given the potential difference (\(V\)) between the two concentric metal spheres as \(900 \mathrm{V}\). We also know the charge (\(Q\)) applied to them is \(6.726 \cdot 10^{-8} \mathrm{C}\) and the radius of the outer sphere (\(R_2\)) is \(0.210 \mathrm{m}\). We need to find the radius of the inner sphere (\(R_1\)).
02

Use the potential difference formula

In this case, we will use the following formula to relate the potential difference, charge, and radii of the spheres: $$ V = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{\frac{1}{R_1} - \frac{1}{R_2}} $$ Where \(\epsilon_0\) is the vacuum permittivity constant and is equal to \(8.854 \times 10^{-12} \mathrm{F/m}\).
03

Rearrange the formula to solve for \(R_1\)

We want to find \(R_1\), so we will rearrange the given formula to solve for it: $$ \frac{1}{R_1} - \frac{1}{R_2} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{V} $$ $$ \frac{1}{R_1} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{V} + \frac{1}{R_2} $$ $$ R_1 = \frac{1}{\frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{V} + \frac{1}{R_2}} $$
04

Calculate the radius of the inner sphere

Now we can plug in the values we are given to find \(R_1\): $$ R_1 = \frac{1}{\frac{1}{4 \pi (8.854 \times 10^{-12} \mathrm{F/m})} \cdot \frac{6.726 \times 10^{-8} \mathrm{C}}{900 \mathrm{V}} + \frac{1}{0.210 \mathrm{m}}} $$ Compute the value of \(R_1\): $$ R_1 \approx 0.141 \mathrm{m} $$ Therefore, the radius of the inner sphere is approximately \(0.141 \mathrm{m}\).

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Most popular questions from this chapter

An \(8.00-\mu F\) capacitor is fully charged by a \(240 .-V\) battery, which is then disconnected. Next, the capacitor is connected to an initially uncharged capacitor of capacitance \(C,\) and the potential difference across it is found to be \(80.0 \mathrm{~V}\) What is \(C ?\) How much energy ends up being stored in the second capacitor?

Two parallel plate capacitors, \(C_{1}\) and \(C_{2},\) are connected in series to a \(96.0-\mathrm{V}\) battery. Both capacitors have plates with an area of \(1.00 \mathrm{~cm}^{2}\) and a separation of \(0.100 \mathrm{~mm} ;\) \(C_{1}\) has air between its plates, and \(C_{2}\) has that space filled with porcelain (dielectric constant of 7.0 and dielectric strength of \(5.70 \mathrm{kV} / \mathrm{mm}\) ). a) After charging, what are the charges on each capacitor? b) What is the total energy stored in the two capacitors? c) What is the electric field between the plates of \(C_{2} ?\)

Fifty parallel plate capacitors are connected in series. The distance between the plates is \(d\) for the first capacitor, \(2 d\) for the second capacitor, \(3 d\) for the third capacitor, and so on. The area of the plates is the same for all the capacitors. Express the equivalent capacitance of the whole set in terms of \(C_{1}\) (the capacitance of the first capacitor).

The potential difference across two capacitors in series is \(120 . \mathrm{V}\). The capacitances are \(C_{1}=1.00 \cdot 10^{3} \mu \mathrm{F}\) and \(C_{2}=1.50 \cdot 10^{3} \mu \mathrm{F}\) a) What is the total capacitance of this pair of capacitors? b) What is the charge on each capacitor? c) What is the potential difference across each capacitor? d) What is the total energy stored by the capacitors?

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?
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