Circuit Elements
Read: Chapter 20
Capacitors
capacitor circuit symbol
A capacitor, in its simplest form, consists of two conducting plates separated by a
small gap. Current cannot flow through a capacitor. A capacitor is capable of storing
charge, so can have a potential difference across its plates.
A fully charged capacitor may be discharged by connecting it to a closed circuit. As
~ decreases in the circuit, causing current to decrease.
charge leaves the capacitor, E
~ = 0 and current stops.
Once the capacitor is fully discharged, E
~
E
−
~
E
+
~
dE
dt
<0
bulb
Question: Using the analysis from Chapter 19, explain why the electric field decreases throughout the circuit as the capacitor
discharges.
1
Circuit Elements
Capacitors
A capacitor may be charged by connecting it to a source of emf (e.g., a battery). As
~ decreases throughout the circuit until the capacitor
charge collects on the capacitor, E
~ and I drop to 0.
becomes fully charged and E
~
E
−
~
E
+
~
dE
dt
−
<0
+
battery
~ in the circuit decreases as the capacitor charges.
Question: Use the analysis of Chapter 19 to explain why E
The rate of charging (or discharging) of a capacitor is dependent upon the current in the circuit — large current→fast
charging, small current→slow charging.
Exercise: Give a physical explanation for the dependence of charging rate on current magnitude.
2
Circuit Elements
Capacitors
In analyzing current in a circuit with a capacitor, the capacitor may be thought of as a single current node — the current
entering one plate must be equal in magnitude to the current leaving the other.
Capacitance C is a the physical property of a capacitor that determines how much charge the capacitor can store at a given
potential difference ∆V ; it is defined by
Q = C∆V .
The value of capacitance is primarily determined by the geometry of the capacitor; for a parallel-plate capacitor, capacitance
is given by C = 0 (A/s), where A is the surface area of a single plate and s is the separation distance between plates.
3
Circuit Elements
Capacitors
In circuit analysis, it is convenient to sometimes define an equivalent capacitance Ceq . Given some configuration of
multiple capacitors, its equivalent capacitance is the value of C that a single capacitor would have if substituted for the
configuration.
V
For capacitors connected in series, we can calculate an equivalent capacitance
Ceq that is defined by
1
1
1
=
+
+ ··· .
Ceq
C1
C2
C1
C2
For capacitors connected in parallel, the equivalent capacitance Ceq is defined
by
V
C1
C2
Ceq = C1 + C2 + · · · .
4
Circuit Elements
Resistors
Conductivity σ is a material property that determines the size of the current in the material at a particular value of applied
~ It is defined by
electric field E.
σ ≡ |q|nu ,
where n is the density of charge carriers and u is their mobility.
Current density J is obtained by dividing current I by the cross-sectional area A. Using I = |q|nuAE, J is defined by
~.
J~ ≡ σ E
Resistance R is a physical property that combines material and geometric elements. The definition of resistance is
L
R≡
,
σA
where L is the length of the circuit element and A its cross-sectional area. The unit of resistance is “ohms:” Ω = V /A.
5
Circuit Elements
Resistors
Ohm’s Law is given by
∆V = IR ,
and is valid for materials whose resistance R is independent of current. Materials that obey Ohm’s Law are said to be ohmic
materials.
Question: Why are capacitors and batteries not ohmic?
resistor circuit element
A resistor is a circuit element that obeys Ohm’s Law. Resistors may be used to
control voltage or current within a circuit.
6
Circuit Elements
Resistors
As with capacitors, we may define an equivalent resitance Req for some configuration of multiple resistors.
For resistors connected in series, equivalent resistance Req is defined by
V
Req = R1 + R2 + · · · .
R1
R2
For resistors connected in parallel, the equivalent resistance Req is defined by
V
R1
R2
1
1
1
=
+
+ ··· .
Req
R1
R2
7
Circuit Elements
Work and Power
The power consumed in any circuit component is given by
P = I∆V .
Exercise:
1.5 V
(a) Calculate the number of electrons that flow past A each second in the
steady-state.
1.5 V
B
A
(b) Draw a graph of potential vs. circuit location, using the labeled points as
points on the x-axis.
(c) What is the power output of the battery?
D
8
40 Ω
C
10 Ω
B
Circuit Elements
Batteries
emf
rint
A real battery may be modeled as an ideal battery coupled to an internal resistance
rint : ∆Vbattery = emf − rint I.
Question: Without internal resistance, what would the current be in a circuit consisting only of a battery and an ideal
(R = 0) wire?
Question: The internal resistance of a battery places an upper limit on what physical quantity?
9
Circuit Elements
RC Circuits
A simple RC circuit consists of a battery, a resistor, and a capacitor connected
in series. The loop equation for the simple RC circuit is
R
emf − IR − Q/C = 0
I
emf
−Q +Q
C
In its steady state, the RC circuit has If = 0 and the capacitor is fully charged with charge Q = (emf)C.
Most generally, the RC circuit has
emf − Q(t)/C
.
R
To find the charge Q on the capacitor at some arbitrary time t, we must solve the differential equation
I(t) =
I(t) =
emf − Q(t)/C
dQ
=
.
dt
R
10
Circuit Elements
RC Circuits
The solution to the differential equation for Q(t) is
−t/RC
Q(coulombs)
Q∼1−e
Q(t) = C(emf)(1 − e−t/RC ) ,
plotted at left in arbitrary units. As t → ∞, Q → C(emf) as expected.
t(seconds)
I(amperes)
I(t) can be obtained by differentiation of Q(t), since I = dQ/dt. Doing so, we
obtain
emf
e−t/RC ,
I(t) =
R
again plotted in arbitrary units. As t → ∞, I → 0 as expected.
I ∼ e−t/RC
t(seconds)
11
Circuit Elements
The time constant τ is defined as the time it takes for the exponential term to
reach e−1 ; for RC circuits, τ = RC. The time constant provides a rough guide
to the rate at which exponential processes evolve. At right are some integer
values of τ on the I vs. t plot.
I(amperes)
RC Circuits
t=τ
t = 2τ
t(seconds)
12
I ∼ e−t/RC