Resource Lesson
A Comparison of RC and RL Circuits
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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.
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Maxwell's Equations
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Meters: Current-Carrying Coils
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Motional EMF
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Parallel Plate Capacitors
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RC Time Constants
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RL Circuits
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Shells and Conductors
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Spherical, Parallel Plate, and Cylindrical Capacitors
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Torque on a Current-Carrying Loop
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