direct current circuits - physics€¦ · the constant ! is called the resistivity of the material....

Direct Current Circuits 1

Direct Current Circuits


Electric Current (Charges in Motion)

Electric Current (I)The net amount of charge that passes through aconductor per unit time at any point. Currentis defined as:

!

I =dQdt

Electric current is measured in coulomb’s per second oramperes. (1A = 1 C/s)In a single circuit, the current at any instant is the sameat one point as any other point. (Charge is conserved.)


The free electrons in conductors are in constantrandom motion, so there is no net flow of chargein any one particular direction.

If a steady electric field is established inside aconductor then charged particles are subjected toa steady force .The charges moving in a conductor have frequentcollisions with the massive nearly stationary ionsof the material.

Drift Velocity

!

v E

!

v F = q

v E


The net effect of the electric field is that inaddition to the random motion of the chargedparticles within a conductor, there is also a verysmall net motion or drift of the moving chargedparticles as a group in the direction of the electricforce.

This motion is described in terms of the driftvelocity of the particles.

Drift Velocity

!

v v d


Drift Velocity

!

"Q = NevdA"t

!

I ="Q"t

!

= NevdA

!

N = concentration of charges m"3( )


Current Density

The current per unit cross-section area is called thecurrent density J.

!

J =IA

!

=NevdAA

!

J = NevdThe electric field E in terms of current density J is:

!

v E = "

v J

The constant ! is called the resistivity of the material.


Batteries (emf)

In order to produce an electric current in a circuit, apotential difference is needed. Batteries are one way ofproviding a difference in potential (called electromotiveforce or emf).

_+

"


Schematic Diagrams

The direction of current is by convention thedirection a positive charge moves through thecircuit, which is towards the negative terminalof the battery.

+#

R

I I

II

II

"


Ohm’s LawGeorg Ohm (1787-1854)• Found that the current in a metal wire is proportional

to the potential difference V applied to its ends.• Current also depends upon the conductivity of the

wire.• It is more common to talk about resistance (inverse of

conductivity) and express this relationship as:

!

I =VR

• The unit for resistance is called the ohm and isabbreviated $ (omega)

!

or V = I " R


Ohm’s Law

+#

For power sources:

ab

I

!

"V =Va #Vb

!

=Vab > 0

For resistive loads:

+ #ab

I

!

"VR =Va #Vb

!

=Vab < 0

R"


Terminal VoltageThe potential difference in a real battery is not equal to the emfdue to internal resistance within the battery. This lowers thevoltage available to the circuit.

Vab is called the terminal voltage of the battery.

_ +

I

r

b a

!

Vab ="# Ir"


Resistance (R) and Resistivity (!)It can be experimentally determined that the resistanceof a wire is directly proportional to its length l andinversely proportional to its cross-sectional area A.

!

R = "l

A

The proportionality constant ! is called the resistivityand depends upon the material used for the wire.

!

" [=] # $m


Electric PowerTo find the power transformed by an electric devicerecall that energy is Q%V. Power is the rate energy istransformed in the device or:

!

P =Q"Vt

The SI unit for power is a J/s or watt (1 W = 1 J/s).For resistors, combining the above with Ohm’s Lawresults in:

!

P = I2R

!

="V 2

R

!

= I"V


Measuring Voltage

• Voltmeters are placed in parallel with the pointsbetween which the voltage measurement is made

• Voltmeters have a very high resistance and do notaffect the circuit (they draw a very smallcurrent)

_+R1

R2

R3

!

Vmeter =VR3V

"


Measuring Voltage

• Voltmeters placed in series will block the currentand create an open circuit.

_+R1

R2

R3

!

Vmeter = E

V

"I = 0

_+R1

R2

R3


Measuring Current

• Ammeters are placed in series with the devicethrough which the current measurement is made

• Ammeters have a very low resistance and do notaffect the circuit (the voltage drop is very low)

_+

R2R1

!

Imeter = IR2A

"


Measuring Current

• Ammeters placed in parallel will create a short circuit.Anything parallel to the meter will have no voltage acrossthem and therefore no current

_+

R2R1

"

AI1 = I2 = 0

R3

!

I =ER3


Resistors in Series

R1 R2 R3

I

ab

!

Vab =V1 +V2 +V3

!

Vab = I R1 + R2 + R3( )

!

I =Vab

R1 + R2 + R3

!

=VabRs

!

= IR1 + IR2 + IR3


Resistors in Series

• Current is the same through each resistor and isthe same as the current in the equivalentresistance

• Voltage drop across each resistor is differentunless the resistance is the same

!

Rs = Rii"

R1 R2 R3

I


Resistors in Series (Voltage Divider)

R1 R2

I

ab

!

I =Vab

R1 + R2

!

V1 = IR1

!

V2 = IR2

!

=VabR2R1 + R2

!

=VabR1R1 + R2


Resistors in Parallel

R1

R2

R3

ab

I

!

I = I1 + I2 + I3

I3

I2

I1

!

1Rp

=1R1

+1R2

+1R3

For resistors in parallel the equivalent resistance is

!

=VabRp

!

=VabR1

+VabR2

+VabR3


Resistors in Parallel

• Voltage drop is the same across each resistorand the same as the voltage drop across theequivalent resistance

• Current is different through each resistor, thehigher the resistance the lower the current

!

1Rp

=1Rii

"

R1

R2

R3


Resistors in Parallel (Current Divider)

R1

R2

ab

II2

I1

!

1R

=1R1

+1R2

!

R =R1R2R1 + R2

!

Vab = I " R1R2R1 + R2

#

$ %

&

' (

!

= I " R2R1 + R2

#

$ %

&

' (

!

I2 =VabR2

so

and

and

!

= I " R1R1 + R2

#

$ %

&

' (

!

I1 =VabR1

!

=R2 + R1R1R2


Kirchhoff’s Rules

1.) Junction Rule (Conservation of charge)At any junction point, the sum of all currentsentering the junction must equal the sum of allcurrents leaving the junction.

i1

!

i1 = i2 + i3i2

i3


Kirchhoff’s Rules

2.) Loop Rule (Conservation of energy)The sum of the changes in potential around any closed path of a circuit is zero.

_+R1

R2

R3

!

" +VR1 +VR2 +VR3 = 0

"


Galvanometers• A galvanometer is simply a meter that deflects

in proportion to the current running through it.• The maximum deflection is called the full-scale

deflection.• The key characteristics of a galvanometer are

– The current Ifs required for full-scale deflection.– The resistance Rc of the coil of wire in the meter.

G

!

V = IRc Rc

I


VoltmetersA voltmeter consists of a resistor put in series witha galvanometer. The value of this resistance Rsdetermines the full-scale reading of the meter Vv.

!

Vv = I fs Rs + Rc( )

Rc

Rs

IfsG

!

Rs =Vv " I fs Rc( )

I fsDirect Current Circuits 28

AmmetersAn ammeter consists of a resistor put in parallel(called a shunt resistor or shunt) with a galvanometer.The value of this resistance Rsh determines the full-scale reading of the meter Ia.

!

I fs Rc = Ia " I fs( )Rsh

Rc

Rsh

Ifs

G

!

Rsh =I fs RcIa " I fs( )

Ia


RC Circuits


RC Circuits (Charging)

!

"# iR # qC

= 0

!

i ="R#qRC

_+

R Cb ca

"

Qo = 0

!

"# vab # vbc = 0

_+

R Cb ca

i +q -q

i

"

+ _


!

i =ER"qRC

!

dqdt

=ER"qRC

!

"dtRC

=dq

q"CE


!

dqdt

=CE " qRC

!

dqdt

= "q"CERC

!

dqq"CE

= "dtRC

!

"dtRC0

t# =

dqq"CE0

q#



!

"tRC

= ln q"CE"CE

# $ %

& ' (

!

q(t) =Qf 1" e" tRC( ) where Qf = C #E

!

e"tRC =

q"CE"CE

!

"tRC 0

t= ln q"CE( ) 0

q

!

"CEe" tRC = q"CE

!

q = CE "CEe" tRC

!

"tRC

" 0 = ln q"CE( )" ln "CE( )

!

"dtRC0

t# =

dqq"CE0

q#



!

i =dqdt

The current at any time is given by:!

q(t ) = Qf 1" e" tRC( )

The charge on the capacitor varies according to:

!

= Ioe"tRC

!

="Re#tRC

RC is called the time constant (& ) and is the time ittakes the capacitor to become 63.2% charged.

!

= C" 1# e# tRC( )



!

q(t ) =Q f 1" e" tRC( )

!

i = Ioe" tRC

t

q

Qf

t

i

Io

RC

Io /e



t

q

Qf

Increasing RC


RC Circuits (Discharging)

!

= "dqdt

!

qC" iR = 0

R Cb ca

+Qo -Qoi = 0

!

vC " vR = 0

!

i =qRC

R Cb ca

i +q -q

i

+_



!

dqqQo

q" = #

dtRC0

t"

!

dqdt

= "qRC

!

ln qQ0

"

# $

%

& ' = (

tRC

!

q =Q0e"tRC

!

ln q( )" ln Q0( ) = "tRC

" 0

!

qQ0

= e"tRC

!

dqq

= "dtRC



!

= "QoRCe"tRC

The current at any time is given by:

The charge on the capacitor varies according to:

!

q =Qoe" tRC

!

i =dqdt

!

= Ioe"tRC



!

i = Ioe" tRC

!

q =Qoe" tRC

t

i

Io

RC

Io /e

t

q

Qo

RC

Qo /e



t

q

Qo

Increasing RC


RC Circuits

!

vC = 0

Uncharged

!

iC " 0shortcircuit

!

vC " 0

!

iC = 0opencircuit

Charged capacitor has same voltage as thedevice that is electrically parallel to it.

!

C

!

C

!

C

!

CFully Charged

direct current circuits - physics€¦ · the constant ! is called the resistivity of the material....

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