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Chapter 21: Problem 18

Show, from \(X_{\mathrm{C}}=1 /(\omega C),\) that the units of capacitive reactance are ohms.

Short Answer

Expert verified
Answer: The units of capacitive reactance are ohms.

Step by step solution

01

Identify the units

Identify the units of each variable in the formula. \(X_C\) represents the capacitive reactance, which is what we want to show has units of ohms. \(\omega\) represents angular frequency and has units of radians per second (rad/s). C represents capacitance and has units of farads (F).
02

Substitute the units

Substitute the units of each variable into the formula: \(X_{\mathrm{C}}=1 /(\omega C)\), and find the resultant units as follows: $$ \mathrm{Ohms} = \frac{1}{\mathrm{rad/s} \cdot \mathrm{F}} $$
03

Replace rad/s with the equivalent expression

Since 1 Hz = 1/s and 1 radian has no units, we can replace \(\omega\) (rad/s) with (1/s). Therefore, the equation becomes: $$ \mathrm{Ohms} = \frac{1}{\mathrm{1/s} \cdot \mathrm{F}} $$
04

Simplify the units

Simplify the units in the denominator, which now becomes seconds multiplied by farads: $$ \mathrm{Ohms} = \frac{1}{\mathrm{s}\cdot \mathrm{F}} $$
05

Invert the denominator

To see that the units are indeed ohms, we can invert the expression in the denominator, which gives: $$ \mathrm{Ohms} = \frac{\mathrm{s}}{\mathrm{F}} $$
06

Recognize the equivalence to ohms

The equation above shows that the units of capacitive reactance are equivalent to the ratio of seconds to farads. This is synonymous with the unit of ohms, hence demonstrating that the units of capacitive reactance are indeed ohms.

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Most popular questions from this chapter

An ac circuit contains a \(12.5-\Omega\) resistor, a \(5.00-\mu \mathrm{F}\) capacitor, and a \(3.60-\mathrm{mH}\) inductor connected in series to an ac generator with an output voltage of \(50.0 \mathrm{V}\) (peak) and frequency of \(1.59 \mathrm{kHz}\). Find the impedance,
A 22 -kV power line that is \(10.0 \mathrm{km}\) long supplies the electric energy to a small town at an average rate of \(6.0 \mathrm{MW} .\) (a) If a pair of aluminum cables of diameter \(9.2 \mathrm{cm}\) are used, what is the average power dissipated in the transmission line? (b) Why is aluminum used rather than a better conductor such as copper or silver?
In an RLC circuit, these three clements are connected in series: a resistor of \(20.0 \Omega,\) a \(35.0-\mathrm{mH}\) inductor, and a 50.0 - \(\mu\) F capacitor. The ac source of the circuit has an rms voltage of \(100.0 \mathrm{V}\) and an angular frequency of $1.0 \times 10^{3} \mathrm{rad} / \mathrm{s} .$ Find (a) the reactances of the capacitor and inductor, (b) the impedance, (c) the rms current, (d) the current amplitude, (e) the phase angle, and (f) the rims voltages across each of the circuit elements. (g) Does the current lead or lag the voltage? (h) Draw a phasor diagram.
At what frequency is the reactance of a \(20.0-\mathrm{mH}\) inductor equal to \(18.8 \Omega ?\)
Suppose that an ideal capacitor and an ideal inductor are connected in series in an ac circuit. (a) What is the phase difference between \(v_{\mathrm{C}}(t)\) and \(v_{\mathrm{L}}(t) ?\) [Hint: since they are in series, the same current \(i(t)\) flows through both. \(]\) (b) If the rms voltages across the capacitor and inductor are \(5.0 \mathrm{V}\) and \(1.0 \mathrm{V}\), respectively, what would an ac voltmeter (which reads rms voltages) connected across the series combination read?
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