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Physics 2113
Jonathan Dowling
Lecture 22: FRI 17 OCT
DC circuits II
Ch27.4-9
How to Solve Multi-Loop Circuits
Monster Maze
If all resistors have a
resistance of 4, and
all batteries are ideal
and have an emf of
4V, what is the
current through R?
i®
Step 1: Find loop that only
goes through R and through
batteries but no other resistors.
Step 2: Assume i is clockwise
on the loop and walk the loop
using the loop rules.
E + E + E - E - iR = 0
Step 3: Solve for i.
2E - iR = 0
i = 2E / R = 2A
Step 4: If i is negative,
reverse the direction of the
current in the diagram.
What If We Chose Wrong
Direction for i ???
If all resistors have a
resistance of 4, and
all batteries are ideal
and have an emf of
4V, what is the
current through R?
i ® ¬i
Step 1: Find loop that only
goes through R and through
batteries but no other resistors.
Step 2: Assume i is counterclockwise on the loop and
walk the loop using the loop
rules.
+E - E - E - E - iR = 0
Step 3: Solve for i.
-2E - iR = 0
i = -2E / R = -2A
Step 4: If i is negative,
reverse the direction of the
current in the diagram.
One Battery? Simplify!
Resistors
Key formula: V=iR
In series: same current dQ/dt
Req=∑Rj
In parallel: same voltage
1/Req= ∑1/Rj
P = iV = i2R = V2/R
Capacitors
Q=CV
same charge Q
1/Ceq= ∑1/Cj
same voltage
Ceq=∑Cj
U = QV/2 = Q2/2C = CV2
Many Batteries? Loop & Junction!
One Battery: Simplify First
Three Batteries: Straight to
Loop & Junction
Apply Loop Rule
Around every loop add +E if you cross a battery from
minus to plus, –E if plus to minus, and –iR for each
resistor. Then sum to Zero: +E1 –E2 – iR1 – iR2 = 0.
R1
E1
+
–
-
-
+
–
R2
Conservation of ENERGY!
E2
Apply Junction Rule
At every junction sum the ingoing currents and outgoing
currents and set them equal.
i1 = i2 + i3
i1
i2
i3
Conservation of CHARGE!
Equations to Unknowns
Continue applying loop and junction until
you have as many equations as unknowns!
Given: E1 , E2 , i1 , R1 , R2
+E1 –E2 – i1R1 – i2 R2 = 0
and
i1 = i2 + i3
Solve for i2 , i3
Example
Find the equivalent resistance between points
(a) F and H and
(b) F and G.
(Hint: For each pair of points, imagine that a battery is
connected across the pair.)
Compile R’s in Series
Compile equivalent R’s in Parallel
Series
H
F
F
H
Parallel
Slide Rule
F
H
Example
Assume the batteries are ideal, and have emf
E1=8V, E2=5V, E3=4V, and R1=140
R2=75and R3=2.
What is the current in each branch?
What is the power delivered by each battery?
Which point is at a higher potential, a or b?
Apply loop rule three times
and junction rule twice.
Example
• What’s the current
through resistor R1?
• What’s the current
through resistor R2?
• What’s the current
through each battery?
Apply loop rule three times
and junction rule twice.
Too many batteries: loop and junction!
There are three loops and two junctions.
Assume all currents are clockwise.
Both junctions give same result:
i1 = i2 + i3
i1 ®
¬ i2
Walk three loops clockwise from a to a:
-E1 - i1R2 + E3 + E2 = 0
¬ i3 ® i3
-8V
i1 = E3 +ER22 -E1 = 4V +5V
= +0.13A
75W
-E1 - i1R2 + E3 - i3 R1 = 0
Finally reverse i3.
5V
= -0.04A
-E2 - i3 R1 = 0 i3 = -E2 / R1 = - 140W
Three equations and three unknowns i1, i2, i3. Tough in general but
trick here is to realize red and green loop equations depend on
only one of the i’s. Solve for i1 and i3 and use junction to get i2.
i2 = i1 - i3 = (+0.13A) - (-0.04A) = +0.17A
i1 ®
E1=8V, E2=5V, E3=4V, and
R1=140R2=75and R3=2.
¬ i2
What is the current in each branch?
i1 = +0.13A i2 = +0.17A i3 = +0.04A
What is the power delivered by
each battery? P = iV = iE
® i3
Which point is at a higher potential, Va = -E1 - i1 R2 + E3 + E2 = 0
a or b?
V = -E - i R < 0
b
Va > Vb
What’s the current through resistor R1
What’s the current through resistor R2?
What’s the current through each battery?
1
1
2
Non-Ideal Batteries
• You have two ideal identical batteries, and a
resistor. Do you connect the batteries in series or
in parallel to get maximum current through R?
• Does the answer change if you have non-ideal (but
still identical) batteries?
Apply loop and junction rules
until you have current in R.
More Light Bulbs
• If all batteries are ideal,
and all batteries and light
bulbs are identical, in
which arrangements will
the light bulbs as bright as
the one in circuit X?
• Does the answer change if
batteries are not ideal?
Calculate i and V across each bulb.
P = iV = “brightness”
or
Calculate each i with R’s the same:
P = i2R
RC Circuits: Charging a Capacitor
In these circuits, current will change for a while, and then stay constant.
We want to solve for current as a function of time i(t)=dq/dt.
The charge on the capacitor will also be a function of time: q(t).
The voltage across the resistor and the capacitor also change with time.
To charge the capacitor, close the switch on a.
E + VR (t) + VC (t) = 0
VC=Q/C
VR=iR
E - i(t)R - q(t) /C = 0
E - ( dq /dt ) R - q(t) /C = 0
A differential equation for q(t)! The solution is:
Time constant:  = RC
q(t) = CE (1- e-t / RC )
Time i drops to 1/e.
-t / RC
® i(t) º dq /dt = (E /R)e
i(t)
E/R
CE
t
t
RC Circuits: Discharging a Capacitor
Assume the switch has been closed
on a for a long time: the capacitor will
be charged with Q=CE.
+++
---
Then, close the switch on b: charges find their way across the circuit,
establishing a current.
V +V = 0
R
C
-i(t)R + q(t) /C = 0 ® ( dq /dt ) R + q(t) /C = 0
+
-C
-t / RC
-t / RC
q(t)
=
q(0)e
=
CEe
Solution:
i(t) = dq /dt = (q(0) /RC)e-t / RC = (E /R)e-t / RC
i(t)
E/R
t
t
(a) Compare i(0) = E / R
i1 (0) > i2 (0) > i3 (0) > i4 (0)
(b) Compare t = RC
t 4 > t 2 = t1 > t 3
Time constant:  = RC
Time i drops to 1/e.
i(t)
E/R
t
Example
The three circuits below are connected to the same ideal
battery with emf E. All resistors have resistance R, and all
capacitors have capacitance C.
•Which capacitor takes the longest in getting charged?
•Which capacitor ends up with the largest charge?
• What’s the final current delivered by each battery?
•What happens when we disconnect the battery?
Simplify R’s into
into Req. Then apply charging
formula with ReqC = 
Example
In the figure, E = 1 kV, C = 10 µF, R1 = R2 = R3 = 1
M. With C completely uncharged, switch S is suddenly
closed (at t = 0).
• What’s the current through each resistor at t=0?
• What’s the current through each resistor after a long
time?
• How long is a long time?
Simplify R1, R2, and R3
into Req. Then apply
discharging formula with
ReqC = 