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Chapter 17: Problem 61

Two metal spheres have charges of equal magnitude, $3.2 \times 10^{-14} \mathrm{C},$ but opposite sign. If the potential difference between the two spheres is \(4.0 \mathrm{mV},\) what is the capacitance? [Hint: The "plates" are not parallel, but the definition of capacitance holds.]

Short Answer

Expert verified
Answer: The capacitance between the two metal spheres is \(8 \times 10^{-12} \mathrm{F}\).

Step by step solution

01

Write down the definition of capacitance

The definition of capacitance is given by the formula: \(C = \frac{Q}{V}\) where C is the capacitance, Q is the charge, and V is the potential difference.
02

Convert the potential difference to volts

The potential difference between the spheres is given as \(4.0 \mathrm{mV}\). To use it in our calculation, we need to convert it to volts. 1 mV = \(10^{-3}\) V So, \(4.0 \mathrm{mV} = 4.0 \times 10^{-3} \mathrm{V}\)
03

Apply the known values

We are given the charge on the spheres which is \(3.2 \times 10^{-14} \mathrm{C}\) and the potential difference converted to volts which is \(4.0 \times 10^{-3} \mathrm{V}\). Substitute the values into the capacitance formula: \(C = \frac{3.2 \times 10^{-14} \mathrm{C}}{4.0 \times 10^{-3} \mathrm{V}}\)
04

Calculate the capacitance

Now, perform the division to find the value of the capacitance: \(C = \frac{3.2 \times 10^{-14}}{4.0 \times 10^{-3}} = 8 \times 10^{-12} \mathrm{F}\) The capacitance of the two metal spheres is \(8 \times 10^{-12} \mathrm{F}\).

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Most popular questions from this chapter

A cell membrane has a surface area of \(1.0 \times 10^{-7} \mathrm{m}^{2},\) a dielectric constant of \(5.2,\) and a thickness of \(7.5 \mathrm{nm}\) The membrane acts like the dielectric in a parallel plate capacitor; a layer of positive ions on the outer surface and a layer of negative ions on the inner surface act as the capacitor plates. The potential difference between the "plates" is \(90.0 \mathrm{mV}\). (a) How much energy is stored in this capacitor? (b) How many positive ions are there on the outside of the membrane? Assume that all the ions are singly charged (charge +e).
(a) If the bottom of a thundercloud has a potential of $-1.00 \times 10^{9} \mathrm{V}\( with respect to Earth and a charge of \)-20.0 \mathrm{C}$ is discharged from the cloud to Earth during a lightning strike, how much electric potential energy is released? (Assume that the system acts like a capacitoras charge flows, the potential difference decreases to zero.) (b) If a tree is struck by the lightning bolt and \(10.0 \%\) of the energy released vaporizes sap in the tree, about how much sap is vaporized? (Assume the sap to have the same latent heat as water.) (c) If \(10.0 \%\) of the energy released from the lightning strike could be stored and used by a homeowner who uses \(400.0 \mathrm{kW}\). hr of electricity per month, for how long could the lightning bolt supply electricity to the home?
A parallel plate capacitor has a charge of \(0.020 \mu \mathrm{C}\) on each plate with a potential difference of \(240 \mathrm{V}\). The parallel plates are separated by 0.40 mm of air. What energy is stored in this capacitor?
The bottom of a thundercloud is at a potential of $-1.00 \times 10^{8} \mathrm{V}\( with respect to Earth's surface. If a charge of \)-25.0 \mathrm{C}$ is transferred to the Earth during a lightning strike, find the electric potential energy released. (Assume that the system acts like a capacitor-as charge flows, the potential difference decreases to zero.)
The potential difference across a cell membrane is \(-90 \mathrm{mV} .\) If the membrane's thickness is \(10 \mathrm{nm},\) what is the magnitude of the electric field in the membrane? Assume the field is uniform.
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