ELECTRIC CURRENT – FLOWING CHARGE (Quick Review from Chap 21)
CURRENT: rate at which charge crosses a specified
surface
• amount of charge per unit time
• surface (for example) could be cross-section of a
wire
+
+
• symbol for current is I
-
∆Q
I
=
Average current is ave ∆t
• ∆Q is amount of charge across surface in time ∆t
Current (I)
DIRECTION OF CURRENT:
• SAME as direction of POSITIVE charges crossing surface
• OPPOSITE direction of NEGATIVE charges crossing surface
INSTANTANEOUS CURRENT (which we will just call current)
I = lim I ave =
∆t → 0
dQ
dt
• Units of current are Amperes: 1 A = 1 Coulomb per second
CHARGE CARRIERS – DEPENDS ON MATERIAL
• In good conductors (metals): current carried by electrons (negative)
o Direction of electron motion opposite to direction of current flow
-e
-e
-e
-e
I (current)
-e
conductor
• In plasma (ionized gas) or electrolyte (i.e. solution in batteries)
o current flow can be carried by:
• positive ions (moving in direction of current flow)
• negative ions (moving opposite to current flow)
• In semiconductors (i.e. in
transistors, computer circuits, etc.)
o current flow can be carried by:
• electrons (negative)
moving in direction
opposite to current flow
• positive charge carriers
(holes) moving in
direction of current flow
-
+
-
+
- +
+
+
- +
+
I (current)
-
-
Material with positive
and negative charge
carriers (i.e. plasma,
electrolyte, or
semiconductor)
MICROSCOPIC PICTURE:
• Look at wire with crosssectional area = A
o Assume number of
mobile charge carriers
per unit volume is n
q
q
vd
vd
A
q
q
vd
WIRE
(area = A)
vd
vd∆t
• Charge carriers have charge q and move right with average speed vd
o Charge carriers move in random directions (thermal motion)
o Undergo collisions (typically 1014/s in Cu) with impurities, imperfections
ƒ After each collision, velocity has random direction
ƒ Speed after collision is thermal speed
o Between collisions, applied electric field causes small additional
acceleration in one direction → average extra velocity is drift velocity
o vd is the magnitude of drift velocity due to field-induced acceleration
between collisions → vd proportional to electric field in wire
MICROSCOPIC PICTURE CONTINUED:
q
q
vd
vd
A
q
q
vd
WIRE
(area = A)
vd
vd∆t
• Volume between two planes separated by distance vd × ∆t is V = vd ∆t A
o In time ∆t , all charge carriers in this volume will cross plane of area A at
right end of this volume
How much charge ∆Q crosses plane, area A in time ∆t ? Ans: ∆Q = n × q × vd ∆t A
∆Q
I
=
= n q vd A
So current is:
∆t
CURRENT THROUGH WIRE: I = n q vd A depends on
• density of charge carriers (good conductors vs. poor conductors)
• the charge of the charge carrier
• drift velocity - set by field in wire and thus by potential difference (voltage)
• area of wire – for given potential difference, bigger wire carries more current
CURRENT DENSITY:
• current per unit cross-sectional area of conductor – does not depend on
shape of wire
J=
I
= n q vd
A
Example:
Charge carriers in a semiconductor have number density n = 3.5 x 1024
carriers/m3. Each carrier has a charge equal to the charge equal to the
charge on an electron.
If the current density is 7.2 x 102 A/m2, what is the speed of the charge
carriers?
CONDUCTIVITY:
I
J
=
= n q vd .
Saw current density is
A
For conductor with electrons (q = -e) for charge carriers, J = −n e vd
• drift velocity is proportional to:
o electric field
o time between collisions (“purity” of material)
• can separate contributions to current density from material properties (carrier
charge, carrier density, time between collisions) and external effect (electric
field E)
Define conductivity σ (depends on material properties only) so that
J = n q vd = σ E
RESISTIVITY:
• resistivity ρ is inverse of conductivity:
ρ=
1
σ
• IMPORTANT: resistivity and conductivity are properties of the material
o They do not depend on the shape of the conducting object
RESISTANCE: relates current through particular object to pot. diff. across it
• Magnitude of the drift velocity depends on:
o electric field and thus on potential difference per unit length in material
• external effect
o time between collisions (purity or physical state of material)
• material property
• Look at wire with cross-sectional area A
o Potential difference between point a
and point b is Vb − Va = ∆V
o Magnitude of electric field in wire is
∆V
E
=
thus
l
o So current is
l
A
a
b
I = JA
=
E
=
A∆V
ρl
ρ
A
ρl
=
R
∆V = IR
o Define resistance (ratio pot. diff. to current) as
A so that
I (current)
Ohm’s Law:
• For materials that obey Ohm’s law: resistance is independent of ∆V
o Such materials (i.e. most metals) are called Ohmic conductors
• For an Ohmic material (i.e. resistance independent of ∆V ), current density
satisfies:
E
J =σ E =
ρ
• Some structures (i.e. diodes) are non-Ohmic
o Resistance depends on potential difference
UNITS:
Resistance unit is Ohm (Ω): 1Ω = 1 V/A
Resistivity unit is ohm-metre (Ω·m)
Conductivity unit is (Ω·m)-1
Example
Calculate the resistance of a length of copper wire 25 cm long and 0.5
mm in diameter. The resistivity of copper is 1.7 × 10−8 Ω ⋅ m .
Example
You have two solid cylinders of the same material. Piece 2 has half the
length and half the diameter of piece 1. What is the ratio of the
resistances of the two pieces?
Example
You have a 100 m long wire of cross-sectional area 0.5 mm2 ,but you do
not know the type of material that makes up the wire.
You have a 12 volt battery and a device to measure current. When the
battery is placed across the two ends of the wire, you measure a current
of 1.07 A.
What is the wire made of? (some sample resistivities are as follows)
Material
Silver
Copper
Gold
Aluminum
Tungsten
Iron
Platinum
Lead
Resistivity (Ω·m)
1.59x10-8
1.7x10-8
2.44 x10-8
2.82 x10-8
5.6 x10-8
10x10-8
11 x10-8
22 x10-8