Bonus section!
Section 3.4 is entitled “Electrical Circuits.”
There are quite a few things you can put into a circuit, but
for the sake of this section, only four of these are important: a voltage
source, an inductor, a capacitor, and a resistor.
Very briefly, a voltage source (think battery or generator) supplies
the voltage to the circuit. Voltage causes electrons to move through the
circuit, and the rate at which these electrons flow is called current. A
resistor limits the current. A capacitor stores charge. An inductor resists the
change in current that passes through it.
If you want more information about this stuff, I recommend
this website, although you could always go to Wikipedia: http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Here’s what a circuit with all four of these items looks
like:
Well, this is what the book portrays these things as. You
would think voltage would be labeled with a V (especially when you consider
that energy is denoted as an E), but sometimes voltage is referred to as the
electromotive force, or emf. So in this case, a voltage source is labeled with
an E. Since the voltage can sometimes be variable (changing) instead of
constant through the circuit, we will denote the voltage source as a function
of time, E(t).
Also, let’s talk about units. The most efficient way of
doing this is to combine everything into one nice-looking chart.
Item
|
Denoted with
|
Units
|
Units denoted with
|
Voltage Source
|
E
|
Volts
|
V
|
Current
|
I
|
Amperes (Amps)
|
A
|
Resistor
|
R
|
Ohms
|
Ω
|
Inductor
|
L
|
Henrys
|
H
|
Capacitor
|
C
|
Farads
|
F
|
Charge
|
Q
|
Coulombs
|
C
|
Perhaps the next thing you’re asking is, “How do we solve a
circuit such as this one?”
My answer will be, “With physics, of course! [And
differential equations. Those too.]”
In order to deal with a circuit such as this one, we need to
have some handy laws and rules and equations to govern ourselves with. The book
calls them component laws, and we’ll look at five of them.
1. Ohm’s Law: The drop across a resistor is proportional to
the current.
2. Faraday’s Law: The voltage drop across
an inductor is proportional to the rate of change of current.
3. Capacitance Law: The voltage drop
across a capacitor is proportional to the charge on the capacitor.
4. Kirchhoff’s voltage law (KVL): The sum
of the voltage drops around any closed loop is zero.
5. Kirchhoff’s current law (KCL): The sum
of currents flowing into a junction equals the sum of the currents flowing out
of that junction (the book says “the sum of currents flowing into any junction
is zero” (129), which is equivalent).
I gave very simple explanations for
beautiful physics equations. If you care, here’s some more information on them
(I especially enjoy Faraday’s Law, but that’s just me).
Capacitance Law: http://www.facstaff.bucknell.edu/mastascu/elessonsHTML/LC/Capac1.htm
(It’s under the section “Voltage-Current Relationships In Capacitors”)
Kirchhoff’s Laws: http://www.facstaff.bucknell.edu/mastascu/elessonshtml/Basic/Basic5Kv.html
and http://www.facstaff.bucknell.edu/mastascu/elessonshtml/Basic/Basic4Ki.html
The beautiful thing about KVL is that
means we can sum the voltage across our three items and the voltage source, and
that resulting voltage will be zero.
The book has the opposite (meaning the
voltage source voltage is negative and the other three are positive), and those
are equivalent. When performing KVL, I tend to think of the circuit having a
total positive voltage E that is coming from the voltage source (since no other
item in our circuit will add any voltage into the circuit). Then I subtract the
voltage that goes across each of the items in our circuit. You can do it the
opposite way, but just make sure you keep track of which is negative and which
is positive.
Something that helps with keeping track
of signs is to sum all the voltages going across the items in your circuit and
setting that equal to the total voltage coming from the voltage source.
We can rewrite the voltages going across
our three items since we have handy laws for each of our three items. Thus
We defined current as the rate at which
electrons (read: charge) flows. This means that I = dQ/dt and we can eliminate
the Q from our equation. By differentiating each side of the equation, we get
Since this is a differential equation and
this is a differential equations blog, we should set some initial conditions to
make our lives easier. Usually, when we start off with a circuit, the charge on
the capacitor is zero (meaning the capacitor is fully discharged) and the
initial current is also zero. However, we still have a tricky second-order
differential equation in our problem. We haven’t formally learned how to solve
a second-order differential equation (spoiler alert: it’s going to be a thing
in the future), so we’re going to save the second order-ness for a different
time (perhaps our bonus section will make a brilliant comeback in the future).
If we just remove the capacitor from the
circuit, then our equation becomes
Now this is something we can solve. It’s
much easier when you already know the resistance and inductance of the circuit.
It’s also much easier when the voltage is a constant rather than a function of
time. In any event, I will leave you with this newfound knowledge of electrical
circuits and bid you adieu until chapter 4.
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