What is equivalent resistance?
Another example
l What are I1, I2 and I3?
Still another example
l What are I1, I2 and I3?
l Let’s start by
simplifying the
circuit?
l What resistors are in
series or parallel with
each other?
l Uh-oh
Kirchoff’s rules
l
l
There are some circuits
that can’t be simplified
any further
To determine currents,
have to apply Kirchoff’s
rules
u
u
this is Kirchoff
these are his rules
1.
2.
sum of currents coming
into a junction equals
sum going out
sum of potential
differences around a
closed loop equals 0
Are these rules hard?
1. Charge is conserved:
1.
2.
for example at c, we can
write I1 + I2 = I3
at b, we can write I3 = I1
+ I2 (but we don’t learn
anything new)
2. Energy is conserved
1.
if I go from a to b to e to f
to c to d, and back to a I
end up at the same place
I started
1.
2.
total change in potential
energy for a charge
making this trip is zero
total change in electric
potential is zero
Applying Kirchoff’s rules
l Let’s see; I have 3
currents (3
unknowns)
l If I remember my
algebra correctly, I’m
going to need 3
equations
l I1 + I2 = I3
u
that’s one
l I3 = I1 + I2
u
doesn’t count
Now I need to apply the loop rule
l Let me start at a and go
to b, c, d and then back
to a
u
+10 V -I1(6 W) - I3(2 W) = 0
u
10 - 6I1 -2I3 = 0
l Now let me go from b to
e to f to c and back to b
u
u
u
-I2(4 W) -14 V + I1(6 W)
-10 V = 0
-4I2 -24 + 6I1 = 0
something tricky here
More rules
l Whenever I travel across a
resistor in the direction of the
current, I lose a potential
equal to -IR
l Whenever I travel across a
resistor against the direction
of the current, I gain a
potential equal to +IR
l Whenever I travel across a
battery from the - to +
terminal, I gain a potential
equal to +e
l Whenever I travel across a
battery from the + to terminal, I lose a potential
equal to -e
Fig. 18.13, p. 564
Answer
l I1 = 2 A
l I2 = -3 A
l I3 = -1 A
l …or
l I2 = +3 A
l I3 = +1 A
l With opposite
directions
Fig. 18.15, p.566
A new element for a circuit
What happens when I close switch S?
l Before S closed, no
current
l After switch is closed,
current starts to flow
l How do we calculate the
current?
l Kirchoff’s loop rule still
applies
u
u
+q
+
-
-q
S around circuit DV = 0
e -q/C -IR = 0
s
s
s
s
at t=0, q=0; I=e/R=Io
at time t, differentiate
equation with respect
to t
dI/dt + I/(RC) = 0
solution: I =Ioe-(t/RC)
+q
RC=‘time constant’
has units of seconds
Fig. 18.16, p.568
Charge on on capacitor as a function of time
at t=0, q=0; at t=infinity, q=Q=Ce
at time t, q=Q(1-e-t/RC)
At time t=RC, q=Q(1-e-1)=0.63Q;
I=Ioe-1=0.37Io
can solve for q or
I at any time t.
Suppose my resistor is a light bulb
l How bright is the light
bulb when I first close
the switch?
l How bright is it if I
wait a ‘long time’?
l What is the power
dissipated in the light
bulb as a function of
time?
u
u
I=Ioe-t/RC
P=I2R=Io2e-2t/RC
What if start with a fully charged capacitor and close the switch?
the charge runs back to the
other plate of the capacitor
I
Q=Qe-t/RC
I=Ioe-t/RC, where Io=Q/RC
suppose my resistor is a
light bulb
Fig. 18.17, p.568
Does Kirchoff’s loop rule hold at all times?
V=e
V across capacitor
V across resistor
time
Example
R=423 W; C=54 mF; e=41 V
What is charge after 2.97E-02 s?