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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 =