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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 RLC series circuit is driven by a sinusoidal emf at the circuit's resonant frequency. (a) What is the phase difference between the voltages across the capacitor and inductor? [Hint: since they are in series, the same current \(i(t) \text { flows through them. }]\) (b) At resonance, the rms current in the circuit is \(120 \mathrm{mA}\). The resistance in the circuit is \(20 \Omega .\) What is the rms value of the applied emf? (c) If the frequency of the emf is changed without changing its rms value, what happens to the rms current? (W) tutorial: resonance)
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Make a figure analogous to Fig. 21.4 for an ideal inductor in an ac circuit. Start by assuming that the voltage across an ideal inductor is \(v_{\mathrm{L}}(t)=V_{\mathrm{L}}\) sin \(\omega t .\) Make a graph showing one cycle of \(v_{\mathrm{L}}(t)\) and \(i(t)\) on the same axes. Then, at each of the times \(t=0, \frac{1}{8} T, \frac{2}{8} T, \ldots, T,\) indicate the direction of the current (or that it is zero), whether the current is increasing, decreasing, or (instantaneously) not changing, and the direction of the induced emf in the inductor (or that it is zero).

A capacitor is connected across the terminals of a 115 \(\mathrm{V}\) rms, \(60.0-\mathrm{Hz}\) generator. For what capacitance is the rms current \(2.3 \mathrm{mA} ?\)
The circuit shown has a source voltage of \(440 \mathrm{V}\) ms, resistance \(R=250 \Omega\), inductance \(L=0.800 \mathrm{H},\) and capacitance $C=2.22 \mu \mathrm{F} .\( (a) Find the angular frequency \)\omega_{0}$ for resonance in this circuit. (b) Draw a phasor diagram for the circuit at resonance. (c) Find these rms voltages measured between various points in the circuit: \(V_{a b}, V_{b c}, V_{c d}, V_{b d},\) and \(V_{c d},\) (d) The resistor is replaced with one of \(R=125 \Omega\). Now what is the angular frequency for resonance? (e) What is the rms current in the circuit operated at resonance with the new resistor?
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