Overview

1
Overview

• Discuss Test 1
• Review
• Kirchoff's Rules for Circuits
• Resistors in Series Parallel
• RC Circuits

Text Reference Chapter 27, 28.1-4
2
Current- a Definition
If there is a potential difference between two
points then, if there is a conducting path, free
charge will flow from the higher to the lower
potential. The amount of charge which flows per
unit time is defined as the current I, i.e.
current is charge flow per unit time.
i.e. no longer electrostatics.
Current
UNIT Ampere A C/s
3
Devices

• Resistors
• Purpose is to limit current drawn in a circuit.
  Resistors are basically bad conductors.
  Actually all conductors have some resistance to
  the flow of charge.

• Resistance
• Resistance is defined to be the ratio of the
  applied voltage to the current passing through.

4
Resistivity

• Property of bulk matter related to resistance

The flow of charge is easier with a larger cross
sectional area, it is harder if L is large.
The resistivity depends on the details of the
atomic structure whichmakes up the resistor (see
chapter 27 in text)
eg, for a copper wire, r 10-8 W-m, 1mm radius,
1 m long, then R .01W
5
Ohm's Law

• Demo
• Vary applied voltage V.
• Measure current I
• Does ratio (V/I) remain constant??

Only true for ideal resisitor!
6
Lecture 11, CQ 1

• Two cylindrical resistors, R1 and R2, are made of
  identical material. R2 has twice the length of R1
  but half the radius of R1.
• These resistors are then connected to a battery V
  as shown

• What is the relation between I1, the current
  flowing in R1 , and I2 , the current flowing in
  R2?

• The resistivity of both resistors is the same
  (r).
• Therefore the resistances are related as

• The resistors have the same voltage across them
  therefore

7
Kirchoff's First Rule"Loop Rule" or Kirchoffs
Voltage Law (KVL)

• "When any closed circuit loop is traversed,
  the algebraic sum of the changes in potential
  must equal zero."

• This is just a restatement of what you already
  know that the potential difference is
  independent of path!

• RULES OF THE ROAD

Move clockwise around circuit
- e1
e2
IR1
IR2
8
Kirchoff's Second Rule"Junction Rule" or
Kirchoffs Current Law (KCL)

• In deriving the formula for the equivalent
  resistance of 2 resistors in parallel, we applied
  Kirchoff's Second Rule (the junction rule).
• "At any junction point in a circuit where the
  current can divide (also called a node), the sum
  of the currents into the node must equal the sum
  of the currents out of the node."

• This is just a statement of the conservation of
  charge at any given node.

9
Resistorsin Series
The Voltage drops
Whenever devices are in SERIES, the current is
the same through both ! This reduces the circuit
to
10
Resistors in Parallel

• What to do?

• Very generally, devices in parallel have the same
  voltage drop

• But current through R1 is not I ! Call it
  I1. Similarly, R2 I2.

• How is I related to I 1 I 2 ?? Current is
  conserved!

11
Loop Demo
12
Lecture 11, CQ 2

• Consider the circuit shown.
• The switch is initially open and the current
  flowing through the bottom resistor is I0.
• Just after the switch is closed, the current
  flowing through the bottom resistor is I1.
• What is the relation between I0 and I1?

13
Lecture 11, CQ 2

• Consider the circuit shown.
• The switch is initially open and the current
  flowing through the bottom resistor is I0.
• After the switch is closed, the current flowing
  through the bottom resistor is I1.
• What is the relation between I0 and I1?

• Write a loop law for original loop

• Write a loop law for the new loop

14
or, Lecture 11, CQ 2

• Consider the circuit shown.
• The switch is initially open and the current
  flowing through the bottom resistor is I0.
• After the switch is closed, the current flowing
  through the bottom resistor is I1.
• What is the relation between I0 and I1?

• The key here is to determine the potential
  (Va-Vb) before the switch is closed.
• From symmetry, (Va-Vb) 12V.
• Therefore, when the switch is closed, NO
  additional current will flow!
• Therefore, the current after the switch is
  closed is equal to the current after the switch
  is closed.

15
Junction Demo
16
RC Circuits
17
Overview of Lecture

• RC Circuit Charging of capacitor through a
  Resistor
• RC Circuit Discharging of capacitor through a
  Resistor

Text Reference Chapter 27.4, 28.2, 28.6
18
RC Circuits
Add a Capacitor to a simple circuit with a
resistor Recall voltage drop on C?
Upon closing circuit Loop rule gives
Recall that
Substituting
Differential Equation for q!
19
Compare with simple resistance circuit

• Simple resistance circuit
• Main Feature Currents are attained
  instantaneously and do not vary with time!!
• Circuit with a capacitor
• KVL yields a differential equation with a term
  proportional to q and a term proportional to I
  dq/dt.

• Physically, whats happening is that the final
  charge cannot be placed on a capacitor instantly.
• Initially, the voltage drop across an uncharged
  capacitor 0 because the charge on it is zero !
• As current starts to flow, charge builds up on
  the capacitor, the voltage drop is proportional
  to this charge and increases it then becomes
  more difficult to add more charge so the current
  slows

20
The differential equation is easy to solve if we
re-write in the form
Integrating both sides we obtain,
Exponentiating both sides we obtain,
If there is no initial charge on C then
21
Charging the Capacitor

22
Lecture 12, CQ 1

• At t0 the switch is thrown from position b to
  position a in the circuit shown The capacitor is
  initially uncharged.
• What is the value of the current I0 just after
  the switch is thrown?

23
Lecture 12, CQ 1

• At t0 the switch is thrown from position b to
  position a in the circuit shown The capacitor is
  initially uncharged.
• What is the value of the current I0 just after
  the switch is thrown?

• Just after the switch is thrown, the capacitor
  still has no charge, therefore the voltage drop
  across the capacitor 0!
• Applying KVL to the loop at t0, IR 0 IR -
  e 0 Þ I e /2R

24
Lecture 12, CQ 1

• At t0 the switch is thrown from position b to
  position a in the circuit shown The capacitor is
  initially uncharged.
• What is the value of the current I0 just after
  the switch is thrown?

(c) I0 2e/R
(b) I0 e/2R
(a) I0 0

• The key here is to realize that as the current
  continues to flow, the charge on the capacitor
  continues to grow.
• As the charge on the capacitor continues to
  grow, the voltage across the capacitor will
  increase.
• The voltage across the capacitor is limited to
  e the current goes to 0.

25
Lecture 12, CQ 2

• At t0 the switch is thrown from position b to
  position a in the circuit shown The capacitor is
  initially uncharged.
• At time tt1t, the charge Q1 on the capacitor is
  (1-1/e) of its asymptotic charge QfCe.
• What is the relation between Q1 and Q2 , the
  charge on the capacitor at time tt22t?

Hint think graphically!
26

• At t0 the switch is thrown from position b to
  position a in the circuit shown The capacitor is
  initially uncharged.
• At time tt1t, the charge Q1 on the capacitor is
  (1-1/e) of its asymptotic charge QfCe.
• What is the relation between Q1 and Q2 , the
  charge on the capacitor at time tt22t?

• So the question is how does this charge
  increase differ from a linear increase?

• From the graph at the right, it is clear that
  the charge increase is not as fast as linear.
• In fact the rate of increase is just
  proportional to the current (dq/dt) which
  decreases with time.
• Therefore, Q2 lt 2Q1.

27
RC Circuits(Time-varying currents)

• Discharge capacitor
• C initially charged with QCe
• Connect switch to b at t0.
• Calculate current and charge as function of time.

Convert to differential equation for q
28
RC Circuits(Time-varying currents)

Check that it is a solution
29
RC Circuits(Time-varying currents)

• Current is found from differentiation

• Conclusion
• Capacitor discharges exponentially with time
  constant t RC
• Current decays from initial max value ( -e/R)
  with same time constant

30
Discharging Capacitor

31
Lecture 12, CQ 3

• At t0 the switch is connected to position a in
  the circuit shown The capacitor is initially
  uncharged.
• At t t0, the switch is thrown from position a
  to position b.
• Which of the following graphs best represents the
  time dependence of the charge on C?

32

• At t0 the switch is connected to position a in
  the circuit shown The capacitor is initially
  uncharged.
• At t t0, the switch is thrown from position a
  to position b.
• Which of the following graphs best represents the
  time dependence of the charge on C?

• For 0 lt t lt t0, the capacitor is charging with
  time constant t RC
• For t gt t0, the capacitor is discharging with
  time constant t 2RC
• (a) has equal charging and discharging time
  constants
• (b) has a larger discharging t than a charging
  t
• (c) has a smaller discharging t than a charging
  t

33
Charging Discharging
RC
2RC
Ce
q
0
t
e/R
I
0
t