Resource Lesson
A Comparison of RC and RL Circuits
Printer Friendly Version
This summary lesson is designed to give students a point-by-point comparison of the properties of inductors and capacitors - two devices frequently encountered in circuits. A classic capacitor (a device designed to stored electric charge) can be represented by a set of parallel plates. A classic inductor (a device which resists changes to the current in a branch of a circuit) can be represented by a solenoid, or coil of wire.
The graph of charge vs voltage shows that capacitance is measured in coulombs/volt which is called a farad (F). We can call this the "operational" definition of capacitance.
The graph of magnetic flux vs current shows that inductance is measured in webers/amp which is called a Henry (H). We can call this the "operational" definition of inductance.
The area under this curve represents the energy stored in the electric field established between the plates.
The area under this curve represents the energy stored in the magnetic field established within the coils.
Using Gauss' Law and the definition of an electric field, we can now define capacitance based on the geometry of the parallel plate capacitor.
Let's begin with a Gaussian "box" whose top is embedded in the top plate, whose sides are parallel to the field lines running from the top to the bottom plate, and whose base is in the electric field within the capacitor.
The only flux lines which penetrate a surface of our box are those that pass through the bottom of the box.
Using Ampere's Law and the definition of magnetic flux, we can define inductance based on the geometry of our inductor, or solenoid.
We will begin with an Amperian loop that runs counterclockwise. The top section (AB) of the loop is outside of the coil's magnetic field, two sides (BC and DA) of the loop are perpendicular to the coil's magnetic fields lines, and the final section (CD) of the loop runs parallel to the field lines.
Notice that capacitance is now based on the area of a plate and the distance between the plates.
e
_{o}
is called of permittivity of free space and represents the "willingness" of the space to establish an electric field. It's value is 8.85 x 10
^{-11}
C
^{2}
/Nm
^{2}
.
Notice that inductance is now based on the coil's geometry: the total number of loops, its total length, and its cross-sectional area. µo is called the permeability of free space and represents the "willingness" of the space to establish a magnetic field. Its value is 4
p
x 10
^{-7}
Tm/A.
Another expression used to describe the electric fields between the plates of the capacitor is the electric field energy density,
u
_{E}
.
Another expression used to describe the magnetic field established within the coils of the solenoid, or inductor, is the magnetic field energy density,
u
_{B}
.
If we look at the graph for charging a capacitor we see that the uncharged capacitor initially acts as if it has "0" resistance to the flow of current. But as charge grows on its plates, it restricts the placement of additional charge (the voltage is fast approaching is maximum value) and the current stops.
If we look at the corresponding graph for an inductor when the switch is initially closed and current starts flowing through the circuit, we see that the inductor acts like it has "infinite" resistance since it opposes (Lenz' Law) any changes in the flux within its coils.
The formula presented for the instantaneous current in the circuit includes a factor called the "RC time constant." Recall that resistance is measured in ohms = volt/amp and that capacitance is measured in Farads = C/volt. If we multiply these two units together we get (volt/amp)(C/volt) = C/amp = seconds. Mathematically this is necessary since e
^{x}
must be a dimensionless value. When one time constant has passed, e
^{-1}
= 0.37, which tells us that only 37% of the maximum current will be remain in the circuit.
The time constant for an inductor-resistor circuit is calculated by L/R. Once again, we can see that the units for the ratio of these variables - t/(L/R) - is dimensionless. L is measured in Henrys = wb/amp. Resistance is measured in ohms = volt/amp. The quotient becomes (wb/amp)/(volt/amp) = wb/volt. A weber is a unit of flux which equals Tm
^{2}
where a tesla = N/(amp m) = N/(Cm/sec).
Thus our time constant becomes [N/(Cm/sec)]m
^{2}
/volt where a volt = J/C = Nm/C.
Substituting in all values yields a final unit of seconds. Our graph shows us that after on time contant, the current will have grown by (1 - 0.37) or 63%.
Once the currents in the circuit reach steady-state conditions, the capacitor is completely charged and behaves like a resistor with "infinite" resistance. No current will flow through the branch of the circuit where the capacitor is located.
Once the currents in the circuit reach steady-state conditions, the inductor no longer has the need to resist any changes in the current. so it behaves like a resistor with "0" resistance.
To learn the voltage across a capacitor we know that
V
_{c}
=
^{q}
/
_{C }
where q is the charge on the plates and C is the capacitance. Often the capacitor is wired in parallel with a resistor allowing us to use the voltage drop across the resistor to be the voltage across the capacitor.
We use Faraday's Law
to determine the voltage across an inductor:
Notice that the induced emf depends on the rate at which the current is changing. Under steady state conditions, dI/dt equals 0 and there is no emf induced in the coil.
Related Documents
Lab:
CP -
Series and Parallel Circuits
Labs -
Aluminum Foil Parallel Plate Capacitors
Labs -
Electric Field Mapping
Labs -
Electric Field Mapping 2
Labs -
Forces Between Ceramic Magnets
Labs -
Magnetic Field in a Solenoid
Labs -
Mass of an Electron
Labs -
Parallel and Series Circuits
Labs -
RC Time Constants
Labs -
Resistance and Resistivity
Labs -
Resistance, Gauge, and Resistivity of Copper Wires
Labs -
Telegraph Project
Labs -
Terminal Voltage of a Lantern Battery
Labs -
Wheatstone Bridge
Resource Lesson:
RL -
A Guide to Biot-Savart Law
RL -
A Special Case of Induction
RL -
Ampere's Law
RL -
An Introduction to DC Circuits
RL -
Capacitors and Dielectrics
RL -
Continuous Charge Distributions: Charged Rods and Rings
RL -
Continuous Charge Distributions: Electric Potential
RL -
Coulomb's Law: Beyond the Fundamentals
RL -
Coulomb's Law: Suspended Spheres
RL -
Derivation of Bohr's Model for the Hydrogen Spectrum
RL -
Dielectrics: Beyond the Fundamentals
RL -
Eddy Currents plus a Lab Simulation
RL -
Electric Field Strength vs Electric Potential
RL -
Electric Fields: Parallel Plates
RL -
Electric Fields: Point Charges
RL -
Electric Potential Energy: Point Charges
RL -
Electric Potential: Point Charges
RL -
Electricity and Magnetism Background
RL -
Electrostatics Fundamentals
RL -
Famous Experiments: Cathode Rays
RL -
Famous Experiments: Millikan's Oil Drop
RL -
Filaments
RL -
Gauss' Law
RL -
Generators, Motors, Transformers
RL -
Induced Electric Fields
RL -
Induced EMF
RL -
Inductors
RL -
Introduction to Magnetism
RL -
Kirchhoff's Laws: Analyzing Circuits with Two or More Batteries
RL -
Kirchhoff's Laws: Analyzing DC Circuits with Capacitors
RL -
LC Circuit
RL -
Magnetic Field Along the Axis of a Current Loop
RL -
Magnetic Forces on Particles (Part II)
RL -
Magnetism: Current-Carrying Wires
RL -
Maxwell's Equations
RL -
Meters: Current-Carrying Coils
RL -
Motional EMF
RL -
Parallel Plate Capacitors
RL -
RC Time Constants
RL -
RL Circuits
RL -
Shells and Conductors
RL -
Spherical, Parallel Plate, and Cylindrical Capacitors
RL -
Torque on a Current-Carrying Loop
Review:
REV -
Drill: Electrostatics
REV -
Drill: Induction
REV -
Electrostatics Point Charges Review
Worksheet:
APP -
Maggie
APP -
The Birthday Cake
APP -
The Circuit Rider
APP -
The Cycle Shop
APP -
The Electrostatic Induction
APP -
The Tree House
CP -
Coulomb's Law
CP -
DC Currents
CP -
Electric Potential
CP -
Electric Power
CP -
Electrostatics: Induction and Conduction
CP -
Induction
CP -
Magnetism
CP -
Ohm's Law
CP -
Parallel Circuits
CP -
Power Production
CP -
Power Transmission
CP -
RIVP Charts #1
CP -
RIVP Charts #2
CP -
Series Circuits
CP -
Transformers
NT -
Bar Magnets
NT -
Brightness
NT -
Electric Potential vs Electric Potential Energy
NT -
Electrostatic Attraction
NT -
Induction Coils
NT -
Light and Heat
NT -
Lightning
NT -
Magnetic Forces
NT -
Meters and Motors
NT -
Parallel Circuit
NT -
Photoelectric Effect
NT -
Potential
NT -
Series Circuits
NT -
Shock!
NT -
Van de Graaff
NT -
Water Stream
WS -
Capacitors - Connected/Disconnected Batteries
WS -
Charged Projectiles in Uniform Electric Fields
WS -
Combinations of Capacitors
WS -
Coulomb Force Extra Practice
WS -
Coulomb's Law: Some Practice with Proportions
WS -
Electric Field Drill: Point Charges
WS -
Electric Fields: Parallel Plates
WS -
Electric Potential Drill: Point Charges
WS -
Electrostatic Forces and Fields: Point Charges
WS -
Electrostatic Vocabulary
WS -
Induced emf
WS -
Introduction to R | I | V | P Charts
WS -
Kirchhoff's Laws: DC Circuits with Capacitors
WS -
Kirchhoff's Laws: Sample Circuit
WS -
Magnetic Forces on Current-Carrying Wires
WS -
Magnetic Forces on Moving Charges
WS -
Parallel Reading - The Atom
WS -
Practice with Ampere's Law
WS -
Practice with Induced Currents (Changing Areas)
WS -
Practice with Induced Currents (Constant Area)
WS -
Resistance, Wattage, and Brightness
WS -
Standard Model: Particles and Forces
TB -
34A: Electric Current
TB -
35A: Series and Parallel
TB -
36A: Magnets, Magnetic Fields, Particles
TB -
36B: Current Carrying Wires
TB -
Advanced Capacitors
TB -
Basic Capacitors
TB -
Basic DC Circuits
TB -
Electric Field Strength vs Electric Potential
TB -
Exercises on Current Carrying Wires
TB -
Multiple-Battery Circuits
TB -
Textbook Set #6: Circuits with Multiple Batteries
PhysicsLAB
Copyright © 1997-2022
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton