Wheatstone Bridge - Department of Physics at UF Chapter 18 34 Wheatstone Bridge ÎAn ammeter A is...

PHY2054: Chapter 18 34

Wheatstone BridgeAn ammeter A is connected between points a and b in the circuit below, in which the four resistors are identical. What is the current through the ammeter?

a) I / 2b) I / 4c) zerod) need more information

The parallel branches have the sameresistance, so equal currents flow ineach branch. Thus (a) and (b) are atthe same potential and there is nocurrent flow across the ammeter.


Circuit Problem (1)The light bulbs in the circuit are identical. What happens when the switch is closed?

(a) both bulbs go out(b) the intensity of both bulbs increases(c) the intensity of both bulbs decreases(d) A gets brighter and B gets dimmer(e) nothing changes

Before: Potential at (a) is 24V, but sois potential at (b) because equalresistance divides 48V in half. When theswitch is closed, nothing will changesince (a) and (b) are still at same potential.


Circuit Problem (2)Bulbs A and B are identical. What happens when the switch is closed?

(a) nothing happens(b) A gets brighter, B dimmer(c) A gets dimmer, B brighter(d) Both A and B get brighter(e) Both A and B get dimmer

Before: Bulb A and bulb B both have 9V across them.

After: Bulb A has 12V across it andbulb B has 6V across it (these potentials are forced by the batteries).


Calculate Currents (Not Simplest Way)

2 1 3

2

1

12 3 06 3 0

i i iii

= +

+ − =+ − =

1

2

2

1

3

424 2 2

iii

=== − =


Res-Monster Maze

All batteries are 4VAll resistors are 4Ω

Find current in R. (Hint: Find voltagesalong path not connected by resistors)


Res-Monster Maze


Find current in R. (Hint: Find voltagesalong path not connected by resistors)

2


Res-Monster Maze (Part 2)


Find currents acrossthese resistors


21 2

011

3

10

11 1

0

1

1 11

1

First find currents in resistors usingpotentials established by batteries

0 0

0


21 2

011

3

10

11 1

0

1

1 11

11

4

1 12

3

8 4677 1

0

1

0 0

0

8

Then use junction rules to find other currents


21 2

011

3

10

11 1

0

1

1 1

111

4

1 12

3

8 4677

9

1

9

8

10

012

11

1 10 9 9

9 0 0

0

Continue using junction rules until finished

1

3 2


Light Bulbs A three-way light bulb contains two filaments that can be connected to the 120 V either individually or in parallel.

A three-way light bulb can produce 50 W, 100 W or 150 W, at the usual household voltage of 120 V.What are the resistances of the filaments that can give the three wattages quoted above?

Use P = V2/RR1 = 1202/50 = 288Ω (50W)R2 = 1202/100 = 144Ω (100W)


ProblemWhat is the maximum number of 100 W light bulbs you can connect in parallel in a 100 V circuit without tripping a 20 A circuit breaker?

(a) 1(b) 5(c) 10(d) 20(e) 100

Each bulb draws a current of 1A. Thus only 20 bulbs are allowed before the circuit breaker is tripped.


Find Currents Through All Resistors

60

Fig. 18-8


Find Currents (2)Find total current (need equivalent R)

10.00Ω + 5.00Ω (parallel) 3.33Ω3.33Ω + 4.00Ω (series) 7.33Ω7.33Ω + 3.00Ω (parallel) 2.13Ω2.13Ω + 3.00Ω (series) 5.13ΩI = 60 / 5.13 = 11.70 A

Current through outside3.00Ω resistor = 11.70 A.

60I


Find Currents (3)Total current I = 11.70, find current I2

One approach, find VA – VB

VB = 0, VA = 60 – 3.00*11.70 = 24.90I2 = 24.9 / 3.00 = 8.30 A

60

I1

I

I2

AB

C


Find Currents (4)Find I1

I1 = I – I2 = 3.40

Find remaining currentsFind ΔV = VA - VC

VC = 0 + 3.40*4.00 = 13.60ΔV = 24.90 – 13.60 = 11.30I10 = 11.30 / 10 = 1.13I5 = 11.30 / 5 = 2.26

60

I1

I

I2

AB

C


Find Currents (5)Find remaining currents

Use I1 = 3.40 Find ΔV = VA - VC

VC = 0 + 3.40*4.00 = 13.60ΔV = 24.90 – 13.60 = 11.30I10 = 11.30 / 10 = 1.13I5 = 11.30 / 5 = 2.26

60

I1

I

I2

AB

C


V

RC CircuitsCharging a capacitor takes time in a real circuit

Resistance allows only a certain amount of current to flowCurrent takes time to charge a capacitor

Assume uncharged capacitor initiallyClose switch at t = 0Initial current is (no charge on capacitor)

Current flows, charging capacitorGenerates capacitor potential of q/C

Current decreases continuously as capacitor charges!Goes to 0 when fully charged

/i V R=

resistor /V V q C RiΔ = − =

i


Analysis of RC CircuitsCurrent and charge are related

So can recast previous equation as “differential equation“

General solution is(Check and see!)K = −CV (necessary to make q = 0 at t = 0)

Solve for charge q and current i

/i dq dt=

dq q Vdt RC R

+ =

/t RCq CV Ke−= +

( )/ /1 t RC t RCdq Vq CV e i edt R

− −= − = =

/V q C Ri− =


Charge and Current vs Time(For Initially Uncharged Capacitor)

( ) ( )/0 1 t RCq t q e−= −

( ) /0

t RCi t i e−=


RC ExampleLet V = 12 volts, C = 6μF, R = 100Ω

τ = RC = 600μs

Charge vs timeUsing units of μC and μsec

Current vs time

Using units of amps and μsec

This circuit will fully charge in a few millisec

( ) ( )/ / 6001 72 1t RC tq CV e e− −= − = −

/ / 6000.12t RC tVi e eR

− −= =


Exponential Behaviort = RC is the “characteristic time” of any RC circuit

Only t / RC is meaningful

t = RCCurrent falls to 37% of maximum valueCharge rises to 63% of maximum value

t =2RCCurrent falls to 13.5% of maximum valueCharge rises to 86.5% of maximum value

t =3RCCurrent falls to 5% of maximum valueCharge rises to 95% of maximum value

t =5RCCurrent falls to 0.7% of maximum valueCharge rises to 99.3% of maximum value


Discharging a CapacitorConnect fully charged capacitor to a resistor at t = 0

Solution is

Solve for current i

R

0qiRC

− − =

0dq qdt RC

+ =

/ /0

t RC t RCq q e CVe− −= =

/ /0

t RC t RCdq Vi e i edt R

− −= = =

C

S

q, i fall exponentially


Charge and Current vs Time(For Initially Charged Capacitor)

( ) /0

t RCq t q e−=

( ) /0

t RCi t i e−=


Same RC ExampleC = 6μF, R = 100Ω

τ = RC = 600μs

Charge capacitor to 12 volts, then pull out battery and let the RC circuit discharge

Charge vs timeUsing units of μC and μsec

Current vs time

Using units of amps and μsec

This circuit will fully discharge in a few millisec

/ / 60072t RC tq CVe e− −= =

/ / 6000.12t RC tVi e eR

− −= =

Wheatstone Bridge - Department of Physics at UF Chapter 18 34 Wheatstone Bridge ÎAn ammeter A is...

Documents

Transcript of Wheatstone Bridge - Department of Physics at UF Chapter 18 34 Wheatstone Bridge ÎAn ammeter A is...