DC Circuits: A Hydraulic Analogy

Resistors in Series

• When 2 or more resistors are connected end-to-end, they are said to be in series – Current is the same in each resistor (conservation of charge) – Sum of potential differences across the resistors = total potential difference across the series combination (conservation of energy) – Equivalent resistance of a series combination of

n

(from Ohm’s Law):

R

eq 

R

1 

R

2 

R

3   

R n

resistors

Resistors in Parallel

• If all of the “left” sides of a set of resistors are connected, along with all of the “right” sides, the resistors are in parallel – Potential differences across the resistors are the same –

I



I

1 

I

2 from conservation of charge (see figure below) – Equivalent resistance of

n

resistors connected in parallel (from Ohm’s Law and conservation of charge): 1

R

eq  1

R

1  1

R

2  1

R

3    1

R n

Household Circuits

CQ 1: The circuit shown below has three resistors connected in parallel to a battery. When an additional resistor,

R

4 , is added to the circuit: A) The voltage produced by the battery will increase. B) The voltage produced by the battery will decrease.

C) The current produced by the battery will decrease.

D) The power produced by the battery will increase.

Example Problem #18.5

(a) Find the equivalent resistance between points

a

and

b

in the circuit diagram. (b) Calculate the current in each resistor if a potential difference of 34.0 V is applied between points

a

and

b

. Solution (details given in class): (a) 17.1 W (b)

I

4 =

I

9 = 1.99 A,

I

7 = 1.17 A,

I

10 = 0.818 A

CQ 2: Interactive Example Problem: Fun with Light Bulbs What would happen if one of the light bulbs were removed from the parallel circuit of light bulbs as shown in class?

A) All of the remaining bulbs go out. B) The remaining bulbs remain illuminated, but with larger brightness.

C) The remaining bulbs remain illuminated with the same brightness.

D) The remaining bulbs remain illuminated, but with smaller brightness.

Kirchhoff’s Rules and Complex DC Circuits

• In more complex DC circuits, resistor combinations cannot be reduced to a single equivalent resistor • These complex circuits can be analyzed using rules known as Kirchhoff’s rules: electrical connection) must equal the current that flows out of the same junction (“junction rule”) • Follows from conservation of charge – The sum of the potential differences across all of the circuit elements around any closed loop must be zero (“loop rule”) • Follows from conservation of energy • Think of taking a hike in the mountains, starting and returning at the same spot (sum of all elevation changes must equal zero no matter what path you take)

Kirchhoff’s Rules: Problem-Solving Strategy

• Assign symbols and directions to the currents in all branches of the circuit • When applying the loop rule, choose a direction for traversing the loop and stay consistent in going either clockwise or counterclockwise • Summary of rules used to record voltage drops and rises while applying loop rule: Direction of loop travel (loop traversal is assumed to be from point

a

toward point

b

)

CQ 3: If all the resistors in the circuit pictured below have equal resistances, and the current through resistor A is 4 A, what is the current through resistor F? A) 2 A B) 1 A C) 8 A D) 16 A

Example Problem #18.17

The ammeter shown in the figure reads 2.00 A. Find

I

1 ,

I

2 , and e .

I

3 #1 #2 Solution (details given in class):

I

1 = 0.714 A

I

2 e = 1.29 A = 12.6 V

I

1

Example Problem #18.26

I

2 #1

I

2 #2

I

3 A dead battery is charged by connecting it to the live battery of another car with jumper cables (see figure). Determine the current in the starter and in the dead battery.

Solution (details given in class):

I

(starter) = 1.7  10 2 A

I

(dead battery) = 0.28 A

I

Example Problem #18.56

I

1

I

–

I

1 #1 #2 The resistor

R

in the figure dissipates 20 W of power. Determine the value of

R

.

Solution (details given in class): 20 W or 98 W

RC

Circuits

• Consider a circuit with a resistor and a capacitor in series with a battery: – Capacitor initially uncharged with switch open – When switch is closed, battery begins to charge capacitor plates and current passes through resistor – Charge on capacitor varies with time:

q



Q

 1 

e



t

/

RC

 Charging a capacitor • •

e

= 2.718 … = Euler’s constant (base of natural logarithms)

Q =

maximum charge =

C

e (only reached at

t

=  ) • •

q =

0 at

t =

0 • At any moment, D

V

(capacitor) =

q RC =

constant = t = /

C

D

V

 e  1 

e



t

/

RC

 time constant of the circuit (units of seconds)

RC

resistor, and a switch:

Circuits

• Now consider a circuit with a charged capacitor, a – Before switch is closed,

Q = C

e Discharging a capacitor – After switch is closed, capacitor discharges exponentially with time:

q



Qe



t

/

RC

– Voltage across capacitor also decreases exponentially with time: D

V

 e

e



t

/

RC

RC

Circuits in Neurons

• A simplified model of axon segments incorporates

RC

circuits (

College Physics

, Giambattista

et al

.) – Current flows through conducting axoplasm fluid (modeled as resistor

R

) – Capacitor consists of interstitial and axoplasm conducting fluids as the plates, with the membrane and myelin sheath acting as an insulator – Speed of electrical impulses depend on time constant of “

RC

circuit” • Typical time constant t for humans is about 20 m s • Speed of electrical impulse = length of axon section/ t ≈ 50 m/s

Example Problem #18.34

A series combination of a 12-k W resistor and an unknown capacitor is connected to a 12-V battery. One second after the circuit is completed, the voltage across the capacitor is 10 V. Determine the capacitance of the capacitor.

Solution (details given in class): 47 m F

CQ 4: Interactive Example Problem:

RC

Circuit Analysis When the battery voltage is doubled in the simulation, how much time does it take for the capacitor to subsequently discharge?

A) 0 ms B) 2 ms C) 4 ms D) 8 ms (PHYSLET Physics Exploration 30.6, copyright Pearson Prentice Hall, 2004)

CQ 5: What is the net force on the dipole inside the capacitor if the plates are separated by 1 cm? A) 0 N B) 4 N C) 8 N D) 16 N

Electrical Safety

• Electric currents passing through the body interfere with the operation of muscles and nervous system • Large currents also cause burns due to energy dissipated in tissues • Current levels and their effects: – 1 mA: usually nothing more than unpleasant sensation – > 5 mA: Harmful effects start to occur – 10 – 20 mA: muscle contractions or paralysis – 100 – 300 mA: ventricular fibrillation if near heart • Most of the electrical resistance of the body is due to the skin – Internal fluids are good conductors due to ions – Total resistance across

dry

– Total resistance across

wet

body ~ 10 – 1000 k W body ~ 1 k W or less

Electrical Safety

• A

short circuit

(low-resistance path) may occur between circuitry inside an appliance and the metal on the outside of appliance – Person touching appliance would then have at least one hand at 120 V with respect to ground – If feet are in a wet tub (which makes good electrical contact with grounded water pipes), resistance may be as low as 500 W – Electrical resistance from one damp hand to another is about 1600 W (still bad) – Three-pronged plugs connect appliance casing to ground – GFI outlets interrupt circuit if current  5 mA is detected “leaking” to ground