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Chapter 23: Problem 32

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.

Short Answer

Expert verified
The maximum capacitance attainable is \(6.0 \mu F\). The minimum capacitance is attained by connecting them in series. Two intermediate capacitances can be found by different combinations of series and parallel connections of capacitors.

Step by step solution

01

Find the Maximum Capacitance

Capacitors in parallel add up their capacitance. To get the maximum capacitance, all the capacitors should be connected in parallel. The total capacitance \( C_{max} \) is the sum of the individual capacitances: \n\[ C_{max} = C_1 + C_2 + C_3 \]Substitute the given capacitance values\[ C_{max} = 1.0 \mu F + 2.0 \mu F + 3.0 \mu F = 6.0 \mu F \]
02

Find the Minimum Capacitance

Capacitors in series reduce the total capacitance. To get the minimum capacitance, all the capacitors should be connected in series. The total capacitance \( C_{min} \) is given by the formula:\[ \frac{1}{C_{min}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \]Substitute the given capacitance values\[ \frac{1}{C_{min}} = \frac{1}{1.0 \mu F} + \frac{1}{2.0 \mu F} + \frac{1}{3.0 \mu F} \]Calculate the above equation to get \( C_{min} \)
03

Find the Intermediate Capacitances

To get the intermediate capacitances, we can combine the capacitors in different series-parallel combinations. Two possible combinations are:1. Connect \(1.0 \mu F\) and \(2.0 \mu F\) capacitors in parallel, and \(3.0 \mu F\) in series.2. Connect \(2.0 \mu F\) and \(3.0 \mu F\) capacitors in parallel, and \(1.0 \mu F\) in series.Calculate the total capacitance for both combinations by using the formulas of series and parallel connections.

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Most popular questions from this chapter

A student argues that the total energy associated with the electric field of a charged sphere must be infinite because its field extends throughout an infinite volume. Critique this argument.

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 \(V_{2}=C_{1} V /\left(C_{1}+C_{2}\right)\).

A classical view of the electron pictures it as a purely electric entity, whose Einstein rest mass energy, \(E=m c^{2},\) is the energy stored in its electric field. If the electron were a sphere with charge distributed uniformly over its surface, what radius would it have in order to satisfy this condition? (Note: Your answer, and the picture of the electron as a sphere, aren't consistent with quantum theory.)

You're evaluating a new hire in your company's engineering department. Together you're working on a circuit where a \(0.1-\mu \mathrm{F}\) \(50-\mathrm{V}\) capacitor is in series with a \(0.2-\mu \mathrm{F}, 200-\mathrm{V}\) capacitor. The new engineer claims you can safely put \(250 \mathrm{V}\) across the combination. What do you say?

An uncharged capacitor has parallel plates \(5.0 \mathrm{cm}\) on a side, spaced \(1.2 \mathrm{mm}\) apart. (a) How much work is required to transfer \(7.2 \mu \mathrm{C}\) from one plate to the other? (b) How much work is required to transfer an additional \(7.2 \mu \mathrm{C} ?\)
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