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Capacitors in AC Circuits: Capacitive Reactance and Phasor Diagrams
The voltage across a capacitor lags the current. Due to the phase difference, it is useful to introduce phasors to describe these circuits.
Learning Objectives

Apply Ohm's law to determine the rms current in the circuit containing only a capacitor

Explain the benefits of using a phasor representation
Key Points

When a capacitor is connected to an alternating voltage, the maximum voltage is proportional to the maximum current, but the maximum voltage does not occur at the same time as the maximum current.

If the AC supply is connected to a resistor, then the current and voltage will be proportional to each other. This means that the current and voltage will "peak" at the same time.

The rms current in the circuit containing only a capacitor C is given by another version of Ohm's law to be
$I_{rms} = \frac{V_{rms}}{X_C}$ , where Xc is the capacitive reactance.
Term
Full Text
In the previous Atom on "Resistors in AC Circuits", we introduced an AC power source and studied how resistors behave in AC circuits. There, we used the Ohm's law (V=IR) to derive the relationship between voltage and current in AC circuits. In this and following Atoms, we will generalize the Ohm's law so that we can use it even when we have capacitors and inductors in the circuit. To get there, we will first introduce a very general, pictorial way of representing a sinusoidal wave, using phasor.
Phasor
The key idea in the phasor representation is that a complex, timevarying signal may be represented as the product of a complex number (that is independent of time) and a complex signal (that is dependent on time). Phasors separate the dependencies on A (amplitude),
Capacitors in AC circuits
Earlier in a previous Atom, we studied how the voltage and the current varied with time. If the AC supply is connected to a resistor, then the current and voltage will be proportional to each other. This means that the current and voltage will "peak" at the same time. We say that the current and voltage are in phase.
When a capacitor is connected to an alternating voltage, the maximum voltage is proportional to the maximum current, but the maximum voltage does not occur at the same time as the maximum current. The current has its maximum (it peaks) one quarter of a cycle before the voltage peaks. Engineers say that the "current leads the voltage by 90^{∘}". This is shown in .
For a circuit with a capacitor, the instantaneous value of V/I is not constant.
However, the value of V_{max}/I_{max} is useful, and is called the capacitive reactance (X_{C}) of the component.
Because it is still a voltage divided by a current (like resistance), its unit is the ohm.
The value of X_{C} (C standing for capacitor) depends on its capacitance (C) and the frequency (f) of the alternating current.
The capacitor is affecting the current, having the ability to stop it altogether when fully charged.
Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor.
This is considered to be an effective resistance of the capacitor to AC, and so the rms current I_{rms} in the circuit containing only a capacitor C is given by another version of Ohm's law to be
Phase representation
Since the voltage across a capacitor lags the current, the phasor representing the current and voltage would be give as in .
In the diagram, the arrows rotate in counterclockwise direction at a frequency
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Key Term Reference
 AC
 Appears in this related concepts: Safety Precautions in the Household, Phase Angle and Power Factor, and Transformers
 Component
 Appears in this related concepts: Adding and Subtracting Vectors Using Components, Position, Velocity, and Acceleration as a Function of Time, and Cathode Ray Tube, TV and Computer Monitors, and the Oscilloscope
 DC
 Appears in this related concepts: Resistors in AC Circuits, Impedance, and Discrete Fourier Transform Examples
 Law
 Appears in this related concepts: TwoComponent Forces, Physics and Other Fields, and Models, Theories, and Laws
 Ohm's law
 Appears in this related concepts: Energy Usage, Current and Voltage Measurements in Circuits, and Ohm's Law
 alternating current
 Appears in this related concepts: Education and the Professions, The Second Industrial Revolution, and The Inventions of the Telephone and Electricity
 atom
 Appears in this related concepts: Description of the Hydrogen Atom, Early Ideas about Atoms, and Periods 1 through 3
 capacitance
 Appears in this related concepts: Capacitors with Dielectrics, Capacitance, and Exercises
 capacitor
 Appears in this related concepts: Resistors and Capacitors in Series, ParallelPlate Capacitor, and ParallelPlate Capacitor
 circuit
 Appears in this related concepts: Combinations of Capacitors: Series and Parallel, Microwaves, and Maxwell's Equations
 complex numbers
 Appears in this related concepts: The Wave Function, Addition, Subtraction, and Multiplication, and Complex Numbers
 current
 Appears in this related concepts: Reporting LongTerm Liabilities, The Battery, and Magnetic Force Between Two Parallel Conductors
 diagram
 Appears in this related concepts: Motion Diagrams, Power, and Bohr Orbits
 frequency
 Appears in this related concepts: Wavelength, Freqency in Relation to Speed, Guidelines for Plotting Frequency Distributions, and General Case
 inductor
 Appears in this related concepts: Induced Charge, Energy Stored in a Magnetic Field, and RL Circuits
 instantaneous
 Appears in this related concepts: Rotational Kinetic Energy: Work, Energy, and Power, Polarization, and Instananeous Velocity: A Graphical Interpretation
 magnitude
 Appears in this related concepts: Roundoff Error, Multiplying Vectors by a Scalar, and Components of a Vector
 ohm
 Appears in this related concepts: Poiseuille's Equation and Viscosity, Null Measurements, and B.2 Chapter 2
 phase
 Appears in this related concepts: The Kinetic Molecular Theory of Matter, The Phase of Orbitals, and The Production of Electromagnetic Waves
 phasor
 Appears in this related concepts: Resonance in RLC Circuits, Inductors in AC Circuits: Inductive Reactive and Phasor Diagrams, and Phasors
 plane
 Appears in this related concepts: Shape and Volume, Center of Mass and Translational Motion, and Shape
 power
 Appears in this related concepts: What is Power?, Sources of Power, and Power
 reactance
 resistance
 Appears in this related concepts: Resistors in Parallel, Resisitors in Series, and Ecology of Ecosystems
 resistor
 Appears in this related concepts: Charging a Battery: EMFs in Series and Parallel, The Loop Rule, and Introduction and Importance
 rms current
 Appears in this related concept: RLC Series Circuit: At Large and Small Frequencies; Phasor Diagram
 rms voltage
 sinusoidal
 Appears in this related concepts: Driven Oscillations and Resonance, Sinusoidal Nature of Simple Harmonic Motion, and Root Mean Square Values
 voltage
 Appears in this related concepts: Conductors, Electric Potential Due to a Point Charge, and Principles of Electricity
 wave
 Appears in this related concepts: Waves, Properties of Waves and Light, and Other Forms of Energy
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Source: Boundless. “Capacitors in AC Circuits: Capacitive Reactance and Phasor Diagrams.” Boundless Physics. Boundless, 03 Jul. 2014. Retrieved 24 Mar. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/inductionaccircuitsandelectricaltechnologies22/accircuits162/capacitorsinaccircuitscapacitivereactanceandphasordiagrams5846271/