February 23, 2014 Physics for Scientists & Engineers 2 1

Review: Ohms law, Kirchhoff’s rules

Physics for Scientists & Engineers 2 2

Current !  Electric current i is the net charge passing a given

point in a given time !  The ampere is abbreviated as A and

is given by

!  The current per unit area flowing through a conductor is the current density

!  If the current is constant and perpendicular to a surface, then and we can write an expression for the magnitude of the current density

dqidt

=

1 C1 A1 s

=

iJA

=

!J

Physics for Scientists & Engineers 2 3

Ohm’s law !  The property of a particular device or object that describes it

ability to conduct electric currents is called the resistance, R !  The definition of resistance R is

!  The unit of resistance is the ohm, Ω

VRi

=

1 V1 1 A

Ω =

Physics for Scientists & Engineers 2 4

Resistivity !  The resistance R of a device is given by

!  ρ is resistivity of the material from which the device is constructed !  L is the length of the device !  A is the cross sectional area of the device

LRA

ρ=

February 23, 2014 Physics for Scientists & Engineers 2, Chapter 25 5

Temperature Dependence of Resistivity

!  The resistivity and resistance vary with temperature !  For metals, this dependence on temperature is linear over a

broad range of temperatures !  An empirical relationship for the temperature dependence

of the resistivity of metals is given by

!  where •  ρ is the resistivity at temperature T •  ρ0 is the resistivity at temperature T0

•  α is the temperature coefficient of electric resistivity

ρ− ρ0 = ρ0α T −T0( ) R− R0 = R0α T −T0( )

February 23, 2014 Physics for Scientists & Engineers 2, Chapter 25 6

Energy and Power in Electric Circuits !  Energy in a circuit is given by

or

!  Power is P = dU /dt, thus electric power is

!  i.e. power loss in a resistor is voltage drop across the resistor multiplied by current through the resistor

!  The unit of power is the Watt (W) !  1 W = 1 A V = 1 V2/Ω = 1 A2Ω !  Thus energy is also in units of 1 J = 1 W s

dU = dqΔV dU = idtΔV

P = iΔV = i2R = ΔV( )2

R

Physics for Scientists & Engineers 2 8

Review – Parallel and Series Resistors !  We can replace n parallel resistors with

one equivalent resistor given by

!  We can replace n series resistors with one equivalent resistor given by

1Req

= 1Rii=1

n

∑ or for two resistors: Req =R1R2

Ri + R2

1

n

eq ii

R R=

=∑

Physics for Scientists & Engineers 2 11

Kirchhoff’s Law, Multi-loop Circuits !  One can create multi-loop circuits that cannot be resolved

into simple circuits containing parallel or series resistors. !  To handle these types of circuits, we must apply Kirchhoff’s

Rules.

! i1 ! i2

! i3

! V1 ! V2 ! V3

Physics for Scientists & Engineers 2 12

Kirchhoff’s Junction rule !  The sum of the currents entering a junction

must equal the sum of the currents leaving a junction

!  Charge conservation

! i1 ! i2

! i3

i1=i2+i3

Physics for Scientists & Engineers 2 14

Kirchhoff’s loop rule !  The sum of voltage drops around a

complete circuit loop must sum to zero. !  Energy conservation !  Suppose that this rule was not valid !  we could construct a way around a loop in such a way that

each turn would increase the potential of a charge traveling around the loop

!  we would always increase the energy of this charge, in obvious contradiction to energy conservation

V1 V2 V3

Physics for Scientists & Engineers 2 15

Circuit Analysis Conventions Element Analysis Direction Current Direction Voltage Drop

→ → −iR

← → +iR

→ ← +iR

← ← −iR

→ +Vemf

← −Vemf

→ −Vemf

← +Vemf

! i is the magnitude of the assumed current through the resistor

Example – Battery Charger !  Two ideal batteries provide V1=12V and

V2=6.0V and the resistors have R1=4.0 Ω and R2=8.0 Ω.

(a) What is the current i? (b) What is the power dissipated in R1

and R2?

February 23, 2014 Physics for Scientists & Engineers 2, Chapter 26 17

V1 V2

V1 − iR2 − iR1 −V2 = 0

P1 = (0.5A)2 (4.0Ω) = 1.0W

P2 = (0.5A)2 (8.0Ω) = 2.0W

" i

Physics for Scientists & Engineers 2 19

Review: RC Circuit !  Charging a capacitor:

!  Discharging a capacitor:

!  Time constant:

0

tRCq q e

⎛ ⎞−⎜ ⎟⎝ ⎠=0

tRCqdqi e

dt RC

⎛ ⎞−⎜ ⎟⎝ ⎠⎛ ⎞= = ⎜ ⎟⎝ ⎠

0( ) 1t

q t q e τ−⎛ ⎞

= −⎜ ⎟⎝ ⎠

temf RCVdqi e

dt R

⎛ ⎞−⎜ ⎟⎝ ⎠⎛ ⎞= = ⎜ ⎟

⎝ ⎠

RCτ =