10/2/2011
From last time
Kirchhoff’s Rules
emf
Internal resistance



1. Junction Rule

Terminal voltage ΔV = ε – Ir
Resistors in Series

Conservation of charge

2. Loop Rule

Resistors in parallel

Same voltage drop
1
1
1
1
=
+
+
+…
Req R1 R2 R3


Σ Vloop = 0
A statement of Conservation of
Energy
Even More About the Loop
Rule
Traveling around the loop
from a to b

In (a), the resistor is
traversed in the direction
of the current, the
potential across the
resistor is –IR

In (b), the resistor is
traversed in the direction
opposite of the current,
the potential across the
resistor is +IR
Problem-Solving Strategy
– Kirchhoff’s Rules






Draw the circuit diagram and assign labels
and symbols to all known and unknown
quantities
Assign directions to the currents.
Apply the junction rule to any junction in
the circuit
Apply the loop rule to as many loops as
are needed to solve for the unknowns
Solve the equations simultaneously for the
unknown quantities
Check your answers
Sum of the ΔV across all the
elements around any closed circuit
loop must be zero

More About the Loop Rule

Sum of the currents entering a
junction = the sum of the currents
leaving the junction
Σ Iin = Σ Iin
Same current
Req = R1 + R2 + R3 + …


I1 = I 2 + I 3
In (c), the source of
emf is traversed in the
direction of the emf
(from – to +), the
change in the electric
potential is +ε
In (d), the source of
emf is traversed in the
direction opposite of
the emf (from + to -),
the change in the
electric potential is -ε
Example Problem 18.26

A dead battery is
charged by
connecting it to a
live battery of
another car with
jumper cables.
Determine the
amount of current in
the starter and in
the dead battery.
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10/2/2011
Charging Capacitor in an
RC Circuit
RC Circuits


DC circuits containing
capacitors and resistors,
having time-varying
currents/charges

The charge on the
capacitor varies with
time
q(t) = Qmax(1 – e-t/RC)
When S is closed, the
capacitor starts to charge
Can define a time
constant:
 = RC



The capacitor charges until it
reaches its maximum charge
(Qmax = Cε)

Once the capacitor is fully
charged, I  0



In a circuit with a large time
constant, the capacitor charges
very slowly
The capacitor charges very quickly
if there is a small time constant
After t = 10 , the capacitor is over
99.99% charged
Example Problem 18.33

for the q to increase
from zero to 63.2% (=
1 – e) of its maximum
Discharging Capacitor in
an RC Circuit
Notes on Time Constant

 is the time required
When a charged
capacitor is placed in
the circuit, it can be
discharged



q = Qe-t/RC
The charge decreases
exponentially
At t =  = RC, the
charge decreases to
0.368 Qmax

In other words, in one
time constant, the
capacitor loses 63.2% of
its initial charge
Solution to 18.26
Consider a series RC
circuit for which R =
1.0 MΩ, C = 5 μF,
and ε = 30 V. Find
the charge on the
capacitor 10 s after
the switch is closed.
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10/2/2011
Solution to 18.33
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