PHYS152
Lecture 6
Ch 28 Circuits
Eunil Won
Korea University
Fundamentals of Physics by Eunil Won, Korea University
Work, Energy, and Emf (기전력)
To produce a steady flow of charge, one need a “charge pump”
: we call it an emf (electromotive force) device
ex) battery, electric generator, solar cells
In any time interval dt, a charge dq passes through and we define
the emf as
dW
E=
dq
SI unit of emf : joule per coulomb = volt
emf arrow: from negative terminal toward the positive terminal
Calculating the Current in a Single-Loop Circuit
dW = E dq = Ei dt
2
Ei dt = i R dt
Fundamentals of Physics by Eunil Won, Korea University
and this has to be equal the thermal energy that appears in the resistor
so
E = iR
or
E
i=
R
Calculating the Current in a Single-Loop Circuit
Loop Rule: The algebraic sum of the changes in
potential encountered in a complete traversal of any
loop of a circuit must be zero (=Kirchhoff’s loop rule)
We traverse a complete loop (clockwise):
Va + E − iR = Va
so
For a counterclockwise travel, we get:
E − iR = 0
−E + iR = 0
For more complex circuits we prepare following rules:
Resistance rule: For a move through a resistance in the direction of the current the change
in potential is -iR (in the opposite direction it is +iR)
Emf rule: For a move through an ideal emf device in the direction of the emf arrow, the
change in potential is +E (in the opposite direction it is -E)
Fundamentals of Physics by Eunil Won, Korea University
Other Single-Loop Circuits
Internal Resistance: real battery has internal resistance (r)
applying the loop rule:
E − ir − iR = 0
or
E
i=
R+r
Resistances in Series
applying the loop rule:
E − iR1 − iR2 − iR3 = 0
E
i=
R1 + R2 + R3
Req = R1 + R2 + R3
The extension to n resistances is
(mathematical induction)
Fundamentals of Physics by Eunil Won, Korea University
Req =
n
!
j=1
Rj
Potential Differences
Potential difference
applying the loop rule:
Vb − iR = Va
E
from the previous result of i =
R+r
so
we get
Power, Potential, and emf
Vb − Va = +iR
R
Vb − Va = E
R+r
The net rate of energy transfer (P) from the emf device to the charge carriers is:
P = iV
where V is the potential across the terminals of the emf device
P = i(E − ir) = iE − i2 r
Pemf = iE
Pr = i2 r (internal dissipation rate)
Fundamentals of Physics by Eunil Won, Korea University
(power of emf device)
Multiloop Circuits
Let’s consider the junction d: charge conservations requires
i1 + i3 = i2
Junction Rule: The sum of the currents entering any junction
must be equal to the sum of the currents leaving that junction
(Kirchhoff’s junction rule)
By applying loop rule (counterclockwise) to left-hand loop
By applying loop rule (counterclockwise) to right-hand loop
By applying loop rule (counterclockwise) to big loop
E1 − i1 R1 + i3 R3 = 0
−i3 R3 − i2 R2 − E2 = 0
E1 − i1 R1 − i2 R2 − E2 = 0
note: there are three unknown currents and we need three equations to solve them completely
(The last equation is sum of two from loop rule)
Resistances in Parallel
V
V
V
,
i
=
,
i
=
Definitions of resistance gives:
2
3
R1
R2
R3
!
"
Applying junction rule at point a:
1
1
1
i = i1 + i2 + i3 = V
+
+
R1
R2
R3
i1 =
Fundamentals of Physics by Eunil Won, Korea University
1
1
1
1
=
+
+
Req
R1
R2
R3
n
!
1
1
=
(n resistances in parallel)
Req
R
j
j=1
RC Circuits
Charging a Capacitor
: capacitor is charged when the switch is closed
applying the loop rule to the circuit
since
i=
dq
dt
q
E − iR −
=0
C
q
dq
we get the following differential equation R
+
=E
dt
C
Let’s assume that the general
solution to this has the form
we require
so
t → ∞,
q = qp + Ke
dq
=0
dt
so
qp : a particular solution
K : a constant to be evaluated
−at
qp = CE
(from the initial condition)
q
dq
(from R + = E )
dt
C
q = CE + Ke−at
initial condition:
t = 0,
plugging it into the
differential equation gives
q=0
RCEae
finally we get:
Fundamentals of Physics by Eunil Won, Korea University
gives
−at
0 = CE + K
+ E − Ee
−at
=E
so q = CE − CEe−at
1
a=
RC
q = CE(1 − e−t/RC )
RC Circuits
q = CE(1 − e−t/RC )
The charging process is now fully described by the solution above
The current charging the capacitor can be obtained as the time
derivative of the charge
dq
i=
=
dt
!
E
R
"
The capacitive time constant
e−t/RC
(1Ω x 1 F = 1 s)
τ = RC
Discharging a Capacitor (no emf)
Fundamentals of Physics by Eunil Won, Korea University
q
dq
=0
R +
q = q0 e−t/RC
dt
C
! q "
dq
0
=−
e−t/RC
i=
dt
RC
Summary
Emf:
Loop rule:
Series Resistances
dW
E=
dq
The algebraic sum of the changes in potential
encountered in a complete traversal of any loop of a
circuit must be zero (=Kirchhoff’s loop rule)
Req =
n
!
Rj
(n resistances in series)
j=1
! 1
1
=
Req
Rj
j=1
n
Resistance
RC circuits
Fundamentals of Physics by Eunil Won, Korea University
(n resistances in parallel)
q = CE(1 − e−t/RC )