Direct CurrentWhen the current in a circuit has a constant magnitude and direction, the current is called direct currentBecause the potential difference between the terminals of a battery is constant, the battery produces direct currentThe battery is known as a source of emf

Electromotive ForceThe electromotive force (emf), ε, of a battery is the maximum possible voltage that the battery can provide between its terminals

The emf supplies energy, it does not apply a force

The battery will normally be the source of energy in the circuit

Sample CircuitWe consider the wires to have no resistanceThe positive terminal of the battery is at a higher potential than the negative terminalThere is also an internal resistance in the battery

Internal Battery ResistanceIf the internal resistance is zero, the terminal voltage equals the emfIn a real battery, there is internal resistance, rThe terminal voltage, ΔV = ε - Ir

Active Figure 28.2

(SLIDESHOW MODE ONLY)

EMF, contThe emf is equivalent to the open-circuitvoltage

This is the terminal voltage when no current is in the circuitThis is the voltage labeled on the battery

The actual potential difference between the terminals of the battery depends on the current in the circuit

Load ResistanceThe terminal voltage also equals the voltage across the external resistance

This external resistor is called the load resistanceIn the previous circuit, the load resistance is the external resistorIn general, the load resistance could be any electrical device

PowerThe total power output of the battery is P = IΔV = IεThis power is delivered to the external resistor (I 2 R) and to the internal resistor (I 2 r)P = Iε = I 2 R + I 2 r

Resistors in SeriesWhen two or more resistors are connected end-to-end, they are said to be in seriesFor a series combination of resistors, the currents are the same in all the resistors because the amount of charge that passes through one resistor must also pass through the other resistors in the same time intervalThe potential difference will divide among the resistors such that the sum of the potential differences across the resistors is equal to the total potential difference across the combination

Resistors in Series, contPotentials add

ΔV = IR1 + IR2= I (R1+R2)

Consequence of Conservation of Energy

The equivalent resistance has the same effect on the circuit as the original combination of resistors

Equivalent Resistance –Series

Req = R1 + R2 + R3 + …The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any individual resistanceIf one device in the series circuit creates an open circuit, all devices are inoperative

Equivalent Resistance –Series – An Example

Two resistors are replaced with their equivalent resistance

Active Figure 28.4

(SLIDESHOW MODE ONLY)

Resistors in ParallelThe potential difference across each resistor is the same because each is connected directly across the battery terminalsThe current, I, that enters a point must be equal to the total current leaving that point

I = I 1 + I 2

The currents are generally not the sameConsequence of Conservation of Charge

Equivalent Resistance –Parallel, Examples

Equivalent resistance replaces the two original resistancesHousehold circuits are wired so that electrical devices are connected in parallel

Circuit breakers may be used in series with other circuit elements for safety purposes

Active Figure 28.6

(SLIDESHOW MODE ONLY)

Equivalent Resistance –Parallel

Equivalent Resistance

The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance

The equivalent is always less than the smallest resistor in the group

1 2 3

1 1 1 1eqR R R R

= + + +K

Resistors in Parallel, FinalIn parallel, each device operates independently of the others so that if one is switched off, the others remain onIn parallel, all of the devices operate on the same voltageThe current takes all the paths

The lower resistance will have higher currentsEven very high resistances will have some currents

Combinations ofResistors

The 8.0-Ω and 4.0-Ω resistors are in series and can be replaced with their equivalent, 12.0 ΩThe 6.0-Ω and 3.0-Ω resistors are in parallel and can be replaced with their equivalent, 2.0 ΩThese equivalent resistances are in series and can be replaced with their equivalent resistance, 14.0 Ω

Statement of Kirchhoff’s RulesJunction Rule

The sum of the currents entering any junction must equal the sum of the currents leaving that junction

A statement of Conservation of Charge

Loop RuleThe sum of the potential differences across all the elements around any closed circuit loop must be zero

A statement of Conservation of Energy

More about the Loop RuleTraveling around the loop from a to bIn (a), the resistor is traversed in the direction of the current, the potential across the resistor is – IRIn (b), the resistor is traversed in the direction opposite of the current, the potential across the resistor is is + IR

Loop Rule, finalIn (c), the source of emf is traversed in the direction of the emf (from – to +), and the change in the electric potential is +εIn (d), the source of emf is traversed in the direction opposite of the emf (from + to -), and the change in the electric potential is -ε

Loop Equations from Kirchhoff’s Rules

The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equationYou need as many independent equations as you have unknowns

Kirchhoff’s Rules Equations, final

In order to solve a particular circuit problem, the number of independent equations you need to obtain from the two rules equals the number of unknown currentsAny capacitor acts as an open branch in a circuit

The current in the branch containing the capacitor is zero under steady-state conditions

Problem-Solving Hints –Kirchhoff’s Rules

Draw the circuit diagram and assign labels and symbols to all known and unknown quantities. Assign directions to the currents.

The direction is arbitrary, but you must adhere to the assigned directions when applying Kirchhoff’s rules

Apply the junction rule to any junction in the circuit that provides new relationships among the various currents

EX.

R

0=VLet

Vi 0RV V− =

R

0=VLet

Vi

The sum of the currents entering any node ( junction ) must be equal to that of leaving currents. ( conservation of energy)

1i

2i

3i1 2 3

1 2 3

0 ( )i i ii i i− − =

= +

Kirchhoffs current rule ( node rule )•

0

R

rise drop

R

V VV V

V V iRViR

− =

= =

=

0

0

R

drop drop

V VV V

iR VViR

− − =

− − =

= −

EX.

6 18 4 10 6 4 024 12

1 ( )2

i i i ii

i A

− + − − − − =− = −

=

Resistance in parallel•　

V1R nR

VeqR

Resistance in series•

V

V1R

nR

eqR1 2By loop rule, ..... 0

k kk k

eq kk

V iR iRV iR i R

VR Ri

− − =

= =

= =

∑ ∑

∑

1 2 31 2

By node rule,1..... ....

1 1

k k

eqk

k

keq k

V Vi i i i VR R R

V VRi V R

R R

= + + + = + + =

= =

=

∑

∑

∑

Ri

motion-V iRΔ =

Ri

motion

V iRΔ =+

(27 – 6)

Resistance Rule：For a move through a resistance in the direction of the current, the change in the potential

.For a move through a resistance in the direction opposite to that of the current, the change in the potential

V iRΔ = −

V iRΔ = +

1 2 3node b (1)i i i= + L

1 1 2 3 1 4 0 (2)V i R i R i R− − − = L

2 3 3 2 0 (3)i R i R− = L

1 1

1 4 3 2 2

3 33 2

1 1 1 00 ....

00

i iR R R i V i

i iR R

⎛ ⎞− − ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− − − = ⇒ =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟

⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠− ⎝ ⎠ ⎝ ⎠⎝ ⎠

b 2R1Re c

3R4R1i

2i 3i

a d?321 =iii

V

... ,

//

2

3

3

2321

43211

RR

iiiii

RRRRVi

=+=

++=

i

V

Ex： Sol. 1、By Ohm’s law

Sol. 2、By Kirchhoff’s rulesNode rule：Loop rule：loop aebda

Loop abed：

In general, for n unknowns take number of sum of nodes and loops = n. Solve these n independent equations.

RC CircuitsA direct current circuit may contain capacitors and resistors, the current will vary with timeWhen the circuit is completed, the capacitor starts to chargeThe capacitor continues to charge until it reaches its maximum charge (Q = Cε)Once the capacitor is fully charged, the current in the circuit is zero

Quick Quiz 28.1

In order to maximize the percentage of the power that is delivered from a battery to a device, the internal resistance of the battery should be (a) as low as possible (b) as high as possible(c) The percentage does not depend on the internal resistance.

Answer: (a). Power is delivered to the internal resistance of a battery, so decreasing the internal resistance will decrease this “lost” power and increase the percentage of the power delivered to the device.

Quick Quiz 28.1

Quick Quiz 28.2

In the figure below, imagine positive charges pass first through R1 and then through R2. Compared to the current in R1, the current in R2 is (a) smaller (b) larger (c) the same

Answer: (c). In a series circuit, the current is the same in all resistors in series. Current is not “used up” as charges pass through a resistor.

Quick Quiz 28.2

Quick Quiz 28.4With the switch in this circuit closed (left), there is no current in R2, because the current has an alternate zero-resistance path through the switch. There is current in R1 and this current is measured with the ammeter (a device for measuring current) at the right side of the circuit. If the switch is opened (Fig. 28.5, right), there is current in R2. What happens to the reading on the ammeter when the switch is opened? (a) the reading goes up(b) the reading goes down(c) the reading does not change

Answer: (b). When the switch is opened, resistors R1and R2 are in series, so that the total circuit resistance is larger than when the switch was closed. As a result, the current decreases.

Quick Quiz 28.4

Quick Quiz 28.8

In using Kirchhoff’s rules, you generally assign a separate unknown current to (a) each resistor in the circuit (b) each loop in the circuit (c) each branch in the circuit (d) each battery in the circuit

Answer: (c). A current is assigned to a given branch of a circuit. There may be multiple resistors and batteries in a given branch.

Quick Quiz 28.8

RC CircuitsA direct current circuit may contain capacitors and resistors, the current will vary with timeWhen the circuit is completed, the capacitor starts to chargeThe capacitor continues to charge until it reaches its maximum charge (Q = Cε)Once the capacitor is fully charged, the current in the circuit is zero

Charging an RC CircuitAs the plates are being charged, the potential difference across the capacitor increasesAt the instant the switch is closed, the charge on the capacitor is zeroOnce the maximum charge is reached, the current in the circuit is zero

The potential difference across the capacitor matches that supplied by the battery

Active Figure 28.19

(SLIDESHOW MODE ONLY)

(27 – 15)

The current

If we rearrange the terms we have：

This is an inhomogeneous, first order, linear differential equation with initial condition： . This condition expresses the fact that at t = 0 the capacitor is uncharged.

0qiRC

− − =E 0dq dq qi Rdt dt C

= → − − =E

dq qRdt C

+ = E

(0) 0q =

RCτ =

(27 – 16)

Differential equation：

Initial condition：

Solution：

Here：

dq qRdt C

+ = E

(0) 0q =

( )/1 tq C e τ−= −E

RCτ =

Charging a Capacitor in an RC Circuit

The charge on the capacitor varies with time

q = Cε(1 – e-t/RC) = Q(1 – e-t/RC)τ is the time constant

τ = RC

The current can be found

I( ) t RCεt eR

−=

Time Constant, ChargingThe time constant represents the time required for the charge to increase from zero to 63.2% of its maximumτ has units of timeThe energy stored in the charged capacitor is ½ Qε = ½ Cε2

Active Figure 28.21

(SLIDESHOW MODE ONLY)

Discharging a Capacitor in an RC Circuit

When a charged capacitor is placed in the circuit, it can be discharged

q = Qe-t/RC

The charge decreases exponentially

Discharging CapacitorAt t = τ = RC, the charge decreases to 0.368 Qmax

In other words, in one time constant, the capacitor loses 63.2% of its initial charge

The current can be found

Both charge and current decay exponentially at a rate characterized by τ = RC

( )I t RCdq Qt edt RC

−= = −

GalvanometerA galvanometer is the main component in analog meters for measuring current and voltageDigital meters are in common use

Digital meters operate under different principles

Galvanometer, contA galvanometer consists of a coil of wire mounted so that it is free to rotate on a pivot in a magnetic fieldThe field is provided by permanent magnetsA torque acts on a current in the presence of a magnetic field

Active Figure 28.27

(SLIDESHOW MODE ONLY)

Galvanometer, finalThe torque is proportional to the current

The larger the current, the greater the torqueThe greater the torque, the larger the rotation of the coil before the spring resists enough to stop the rotation

The deflection of a needle attached to the coil is proportional to the currentOnce calibrated, it can be used to measure currents or voltages

AmmeterAn ammeter is a device that measures currentThe ammeter must be connected in series with the elements being measured

The current must pass directly through the ammeter

Ammeter in a CircuitThe ammeter is connected in series with the elements in which the current is to be measuredIdeally, the ammeter should have zero resistance so the current being measured is not altered

Ammeter from GalvanometerThe galvanometer typically has a resistance of 60 ΩTo minimize the resistance, a shunt resistance, Rp, is placed in parallel with the galvanometer

Active Figure 28.29

(SLIDESHOW MODE ONLY)

Ammeter, finalThe value of the shunt resistor must be much less than the resistance of the galvanometer

Remember, the equivalent resistance of resistors in parallel will be less than the smallest resistance

Most of the current will go through the shunt resistance, this is necessary since the full scale deflection of the galvanometer is on the order of 1 mA

VoltmeterA voltmeter is a device that measures potential differenceThe voltmeter must be connected in parallel with the elements being measured

The voltage is the same in parallel

Voltmeter in a CircuitThe voltmeter is connected in parallel with the element in which the potential difference is to be measured

Polarity must be observedIdeally, the voltmeter should have infinite resistance so that no current would pass through it

Voltmeter from GalvanometerThe galvanometer typically has a resistance of 60 ΩTo maximize the resistance, another resistor, Rs, is placed in series with the galvanometer

Voltmeter, finalThe value of the added resistor must be much greater than the resistance of the galvanometer

Remember, the equivalent resistance of resistors in series will be greater than the largest resistance

Most of the current will go through the element being measured, and the galvanometer will not alter the voltage being measured

Household WiringThe utility company distributes electric power to individual homes by a pair of wiresEach house is connected in parallel with these wiresOne wire is the “live wire” and the other wire is the neutral wire connected to ground

Household Wiring, contThe potential of the neutral wire is taken to be zero

Actually, the current and voltage are alternating

The potential difference between the live and neutral wires is about 120 V

Household Wiring, finalA meter is connected in series with the live wire entering the house

This records the household’s consumption of electricity

After the meter, the wire splits so that multiple parallel circuits can be distributed throughout the houseEach circuit has its own circuit breakerFor those applications requiring 240 V, there is a third wire maintained at 120 V below the neutral wire

Short CircuitA short circuit occurs when almost zero resistance exists between two points at different potentialsThis results in a very large currentIn a household circuit, a circuit breaker will open the circuit in the case of an accidental short circuit

This prevents any damageA person in contact with ground can be electrocuted by touching the live wire

Electrical SafetyElectric shock can result in fatal burnsElectric shock can cause the muscles of vital organs (such as the heart) to malfunctionThe degree of damage depends on:

the magnitude of the currentthe length of time it actsthe part of the body touching the live wirethe part of the body in which the current exists

Effects of Various Currents5 mA or less

can cause a sensation of shockgenerally little or no damage

10 mAmuscles contractmay be unable to let go of a live wire

100 mAif passing through the body for 1 second or less, can be fatalparalyzes the respiratory muscles

More EffectsIn some cases, currents of 1 A can produce serious burns

Sometimes these can be fatal burnsNo contact with live wires is considered safe whenever the voltage is greater than 24 V

Ground WireElectrical equipment manufacturers use electrical cords that have a third wire, called a groundThis safety ground normally carries no current and is both grounded and connected to the appliance

Ground Wire, contIf the live wire is accidentally shorted to the casing, most of the current takes the low-resistance path through the appliance to the groundIf it was not properly grounded, anyone in contact with the appliance could be shocked because the body produces a low-resistance path to ground

Ground-Fault Interrupters (GFI)Special power outletsUsed in hazardous areasDesigned to protect people from electrical shockSenses currents (of about 5 mA or greater) leaking to groundShuts off the current when above this level

The constant is known as the “time constant” of the circuit. If we plot q versus t we see that q does not reach its terminal value but instead increases from its initial value and it reaches the terminal value at . Do we have to wait for an intercity to charge the capacitor？ In practice no.

If we wait only a few time constants the charge, for all practical purposes has reached its terminal value .

The current If we plot i versus t we get a decaying exponential (see fig. b)

τ

CEt = ∞

( )( )( )

( ) 0.632

( 3 ) 0.950

( 5 ) 0.993

q t C

q t C

q t C

τ

τ

τ

= =

= =

= =

E

E

E

CE/ tdqi e

dt Rτ−⎛ ⎞= = ⎜ ⎟

⎝ ⎠E

+-

i

q

t

qo

O

RC

(27 – 17)

τ =

RC circuits：Discharging of a capacitorConsider the circuit shown in the figure. We assume that the capacitor at t = 0 has charge and that at t = 0 we throw the switch S from the middle position to position b. The capacitor is disconnected from the battery and looses its charge through resistor R. We will write KLR starting at point b and going in the counterclockwise direction：

oq

0q iRC

− − =

Taking into account that we get：dqidt

= 0dq qRdt C

+ =

This is an homogeneous, first order, linear differential equation with initial condition： . The solution is： where . If we plot q versus t we get a decaying exponential. The charge becomes zero at In practical terms we only have to wait a few time constants.

( ) ( ) ( )( ) 0.368 , (3 ) 0.049 , (5 ) 0.007o o oq q q q q qτ τ τ= = =

(0) oq q=- /t

oq q e τ= RCτ =t = ∞

Ohm’s law1、Which statement (s) is (are) true？When a long

straight conducting wire of constant cross-section is connected to the terminals of a battery. The electric field

___ A、lines are uniformly distributed over the cross-sectional area of the conductor

___ B、inside the wire is of constant magnitude and its direction is parallel to the wire

___ C、both of the above___ D、neither of the above

Quick Quiz 28.9a

Consider the circuit seen below and assume that the battery has no internal resistance. Just after the switch is closed, the potential difference across which of the following is equal to the emf of the battery?

(a) C (b) R (c) neither C nor R.

Quick Quiz 28.10a

Consider the circuit in the figure below and assume that the battery has no internal resistance. Just after the switch is closed, the current in the battery is (a) zero (b) ε/2R (c) 2ε /R (d) ε/R (e) impossible to determine.

Answer: (c). Just after the switch is closed, there is no charge on the capacitor. Current exists in both branches of the circuit as the capacitor begins to charge, so the right half of the circuit is equivalent to two resistances R in parallel for an equivalent resistance of 1/2R.

Quick Quiz 28.10a

Quick Quiz 28.10b

After a very long time, the current in the battery is (a) zero (b) ε/2R (c) 2ε/R (d) ε /R (e) impossible to determine.

Answer: (d). After a long time, the capacitor is fully charged and the current in the right-hand branch drops to zero. Now, current exists only in a resistance R across the battery.

Quick Quiz 28.10b