# Current and Current Density

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```					                         Current and Current Density
•   Consider moving charges
∆q

v∆t                                   I=∆q/∆t

•   current units: Coulombs/second (C/S) = Ampères (A) “Amps”
•   previous weeks, we considered the electrostatic cases (I = 0), where
we were able to say that in every conductor
– E=0
– V = constant
•   now for conductors with non-zero resistivity, we can have
– E ≠0
– V≠ constant

Phy 1223 - Winter 2003                                                                      1
Dr. Bala Maheswaran

Current and Current Density (cont.)
•   Conservation of charge implies conservation of current in the steady
state:
Iout1
Iin1
Iout2
∑I       in   = ∑ I out
Iin2      Iout3

•   Sign convention: current is the net flow of positive charge
I                                                             I
+                                                       -

+                                                         -
I=0
+              I                                             + +
-                                                                  -
Phy 1223 - Winter 2003                                                                      2
Dr. Bala Maheswaran

1
Currents and Current Density
•   Usual case in a conductor: the current results from the drift of
electrons (in direction opposite the current)
•   Counter examples: proton beams, positive ions in solution (positive
charges moving in same direction as current)

•   Current density (J): charge per second per area (A/m2 )

J1 =I/A 1
J2 =I/A 2

I                            I

Phy 1223 - Winter 2003                                                             3
Dr. Bala Maheswaran

Current and Current Density (cont.)
•   Example: Melting point for a fuse material is 440 A/cm2. What should
be the diameter of a cylinder to make a 2A fuse?

•   Inside of a conducting material, the current density is proportional to
the density of charge carriers (n), the charge on each carrier (q) and the
drift velocity of the carriers (v d):
r     r
J = nqvd
•   usually, the carriers are electrons, so n is the free electron density (free
electrons/m3), q is -1.6e-19 C.
•   drift velocity very slow (<mm/s), should not be confused with velocity
of individual electrons

Phy 1223 - Winter 2003                                                             4
Dr. Bala Maheswaran

2
Resistance and Resistivity
•   It turns out that for many materials, the drift velocity is nearly directly
proportional to the electric field (Ohm’s law), so the whole thing can
be reduced to one constant: resistivity (ρ)
•   Resisitivity is a property of a material
– defined by ρ = E/J
– units : Vm/A, or Ohm-meters (Ωm)
– reciprocal: conductivity
•   Resistance is a property of an object
– defined by R=V/I
ρL
– units: V/A, or Ohms (Ω)
R=
V=EL                L           V=0                      A
E                 A
J

Phy 1223 - Winter 2003                                                            5
Dr. Bala Maheswaran

Resistance and Resistivity (cont.)
•   Examples of resistivity:

Silver        1.62x10 -8 Ωm              Aluminum 2.75x10-8 Ωm

Copper        1.69x10 -8 Ωm              Iron        9.68x10-8 Ωm

Silicon       2.5x103 Ωm

•   How much 14-gauge copper wire to make a 5Ω resistor? (diameter =
1.63 mm)
•   longer: higher resistance, fatter: lower resistance (think of water pipes)

Phy 1223 - Winter 2003                                                            6
Dr. Bala Maheswaran

3
Resistance and Resistivity (cont.)
•   Power dissipated in a resistor:
P=IV=I2 R=V2/R
– units: (C/s)(J/C)=J/s=Watts (W)

•   Example: 60W light bulb at 120 V
– I=0.5 A
– R=240 Ω

Phy 1223 - Winter 2003                                                         7
Dr. Bala Maheswaran

EMF, Potential, and Voltage
•   EMF, “Electromotive Force”
– not a force
– units: Volts
– symbol
– used to characterize voltage sources (batteries, generators, etc.)
•   the “voltage” usually reverse to the difference in potential between the
terminals of the EMF source
•   The source tries to maintain a voltage of across its terminals
•   V ba= when no current is flowing
•   if the internal resistance (r ) is small, and V are essentially the same

Vb                r        Va
Vba= -Ir

Vba
Phy 1223 - Winter 2003                                                         8
Dr. Bala Maheswaran

4
Direct Current Circuits
•   Assume wires are perfect conductors: V=constant at every point on a
wire.
•   Algebraic sum of voltages around a loop must be zero
•   current is conserved at each node

_     +

Phy 1223 - Winter 2003                                                    9
Dr. Bala Maheswaran

5

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