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Chapter 23: Problem 44

Which can store more energy: a \(1.0-\mu \mathrm{F}\) capacitor rated at \(250 \mathrm{V}\) or a 470 -pF capacitor rated at \(3 \mathrm{kV} ?\)

Short Answer

Expert verified
The 1.0-\(\mu\)F capacitor rated at 250 V can store more energy than the 470-pF capacitor rated at 3 kV.

Step by step solution

01

Conversion of units

First, convert all units into the standard measurement (Farads for capacitance and volts for voltage). A microfarad \(\mu\mathrm{F}\) is \(10^{-6}\) Farad and a picofarad \(pF\) is \(10^{-12}\) Farad while a kiloVolt \(kV\) is \(10^3\) Volts. Therefore, the first capacitor's capacitance is \(1.0 \times 10^{-6}\) F and its voltage is 250 V. The second capacitor's capacitance is \(470 \times 10^{-12}\) F and its voltage is \(3 \times 10^3\) V.
02

Calculation of energy stored in each capacitor

The energy stored \(U\) in a capacitor is given by the formula \(U = \frac{1}{2} C V^2\). Substitute the values of \(C\) and \(V\) for each capacitor into the equation and solve for \(U\). For the first capacitor \(U_1 = \frac{1}{2}(1.0 \times 10^{-6})(250)^2 = 0.03125\) J and for the second capacitor \(U_2 = \frac{1}{2}(470 \times 10^{-12})(3000)^2 = 0.002115\) J.
03

Comparison of stored energies

The energy stored by the first capacitor is greater than that of the second capacitor. Therefore, the 1.0-\(\mu\)F capacitor rated at 250 V can store more energy than the 470-pF capacitor rated at 3 kV.

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Most popular questions from this chapter

Two capacitors are connected in series and the combination is charged to \(100 \mathrm{V}\). If the voltage across each capacitor is \(50 \mathrm{V}\), how do their capacitances compare?

How does the energy density at a certain distance from a negative point charge compare with the energy density at the same distance from a positive point charge of equal magnitude?

Capacitors \(C_{1}\) and \(C_{2}\) are in series, with voltage \(V\) across the combination. Show that the voltages across the individual capacitors are \(V_{1}=C_{2} V /\left(C_{1}+C_{2}\right)\) and \(V_{2}=C_{1} V /\left(C_{1}+C_{2}\right)\).

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You're given three capacitors: \(1.0 \mu \mathrm{F}, 2.0 \mu \mathrm{F},\) and \(3.0 \mu \mathrm{F} .\) Find (a) the maximum, (b) the minimum, and (c) two intermediate capacitances you could achieve using combinations of all three capacitors.
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