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Chapter 25: Problem 31

Show that the quantity \(R C\) has the units of time (seconds).

Short Answer

Expert verified
The quantity \(R C\) has the units of time (seconds).

Step by step solution

01

Identify the units for each quantity

The units for resistance (\(R\)) is Ohms (惟) and for capacitance (\(C\)) is Farads (F). So, the given quantity \(R C\) signifies that we multiply the resistance by the capacitance.
02

Understand the meaning of Ohm and Farad

An Ohm is defined as volts per ampere (\(V/A\)), and a Farad is defined as Coulomb per volt (\(C/V\)). So when we multiply Ohm (惟) and Farad (F), we actually multiply \(V/A\) and \(C/V\).
03

Multiply the Ohm and Farad units

As explained in the previous step, we are essentially multiplying \(V/A\) with \(C/V\). This equals to \(C/A\), since volts in the numerator (from resistance's units) and volts in the denominator (from capacitance's units) cancel out.
04

Translate back to time unit

The unit \(C/A\) is same as seconds (s), where Coulomb is unit of electric charge which is equivalent to Ampere-second (As) and Ampere is the unit of electric current. Thus, \(C/A = As/A = s\). This means that, indeed, the unit of the quantity \(R C\) is seconds (s).

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Most popular questions from this chapter

Are household electrical outlets connected in series or parallel? How do you know?

Obtain an expression for the rate of increase ( \(d V / d t\) ) of the voltage across a charging capacitor in an \(R C\) circuit. Evaluate your result at time \(t=0,\) and show that if the capacitor continued charging steadily at this rate, it would reach full charge in exactly one time constant.

You have a \(1.0-\Omega,\) a \(2.0-\Omega,\) and a \(3.0-\Omega\) resistor. What equivalent resistances can you form using all three?

The voltage on a charged capacitor is monitored with a voltmeter whose resistance is \(1.00 \mathrm{M} \Omega .\) The table below gives the meter reading as a function of time. Determine a function of the voltage which, when you plot it against time, should give a straight line. Make your plot, establish a best-fit line, and use it to determine the capacitance.$$\begin{array}{|l|c|c|c|c|c|}\hline \text { Time (s) } & 0 & 1 & 2 & 3 & 4 \\\\\hline \text { Voltage (V) } & 15.0 & 10.3 & 6.36 & 3.78 & 2.43 \\\\\hline\end{array}$$

Show that a battery delivers the most power when the load resistance across its terminals is equal to its internal resistance. (This is not the way to treat a battery, but it's the basis for load matching in amplifiers; see Problem \(69 .\) )
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