Transcript Current, Resistance, DC Circuits

Physics 121: Electricity &
Magnetism – Lecture 7
Current & Resistance
Dale E. Gary
Wenda Cao
NJIT Physics Department
Definition of Current
Current is the flow of electrical
charge, i.e. amount of charge per Units: ampere
1 A = 1 C/s
second moving through a wire, i =
dq/dt.
 It is a scalar, not a vector, but it
has a direction—positive in the
direction of flow of positive charge
carriers.
 Any way that you can get charges 11 A - 3 A = 8 A
to move will create a current, but
in
a typical way is to attach a battery Total current out
to a wire loop.
 Charges will flow from the +
terminal to the – terminal (again,
it is really electrons that flow in
the opposite direction, but current
is defined as the direction of
positive charge carriers).

8A
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Current in a Circuit
 What
is the current in the wire marked i in
the figure below?
11 A - 3 A = 8 A
Total current out
in
8A
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Current At Junctions
 What
is the current in all of the wire
sections that are not marked?
5A
3A
2A
6A
8A
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Try One Yourself
1.
What is the current in the wire section
marked i?
1A
A.
B.
C.
D.
E.
1 A.
2A
2 A.
5 A.
5A
7 A.
Cannot determine
from information given.
3A
2A
i
6A
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Current Density
When we care only about the total current i in a conductor, we do not have
to worry about its shape.
 However, sometimes we want to look in more detail at the current flow
inside the conductor. Similar to what we did with Gauss’ Law (electric flux
through a surface), we can consider the flow of charge through a surface.
To do this, we consider (charge per unit time) per unit area, i.e. current per
unit area, or current density. The units are amps/square meter (A/m2).
 Current density is a vector (since it has a flow magnitude and direction). We
J relationship between current and current density is
use the symbol . The

 
i   J  dA
High current density
in this region
Small current density
in this region
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Drift Speed

Let’s look in detail at one happens when we connect a battery to a wire to
start current flowing.
Current
Thermal motions
of electrons—no
net
drift. drift
Electrons
in direction
opposite to i
_
1.5 V
battery
+
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Drift Speed
The drift speed is tiny compared with thermal motions.
6
 Thermal motions (random motions) have speed vth  10 m/s
 Drift speed in copper is 10-4 m/s .
 Let’s relate drift speed to current density.

A
vd
n
L
  dq
i   J  dA 
 neAvd
dt
Total charge q in volume V
N
q  Ve  nVe  nALe
V
density of
charge carriers
L  vd t  q  nAvd te
time to drift
a distance L

 +e means J and vd in same direction
J  nevd
-e means J and vd in opposite directions
ne is carrier charge density r
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Increasing the Current
2.
When you increase the current in a wire,
what happens?
A.
The number of charge carriers stays the same, and the
drift speed increases.
The drift speed stays the same, and the number of
charge carriers increases.
The charge carried by each charge carrier increases.
The current density decreases.
B.
C.
D.
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Resistance
V
Resistance is defined to be R  . That is, we apply a voltage V, and ask
i
how much current i results.
This is called Ohm’s Law.
 If we apply the voltage to a conducting wire, the current will be very large
so R is small.
 If we apply the voltage to a less conducting material, such as glass, the
current will be tiny so R is very large.


The unit of resistance is the ohm, W.
(Greek letter omega.)
1 ohm = 1 W = 1 volt per ampere = 1 V/A
Resistor
V Circuit
Diagram
R
+
i
i small,
large, so
R small
large
1.5 V
battery
_
glass
wire
filament
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Current Through a Resistor
3.
What is the current through the resistor in
the following circuit, if V = 20 V and R =
100 W?
A.
20 mA.
5 mA.
0.2 A.
200 A.
5 A.
B.
C.
D.
E.
V Circuit
Diagram
R
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Current Through a Resistor
4.
If the current is doubled, what changes?
A.
V Circuit
The voltage across the resistor doubles.
Diagram
The resistance of the resistor doubles.
The voltage in the wire between the battery and the
resistor doubles.
The voltage across the resistor drops by a factor of 2.
The resistance of the resistor drops by a factor of 2.
B.
C.
D.
E.
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R
Resistivity and Conductivity
Rather than consider the overall resistance of an object, we can discuss the
property of a material to resist the flow of electric current.
 This is called the resistivity. The text uses (re-uses) the symbol r for
resistivity. Note that this IS NOT related to the charge density, which we
discussed earlier.
 The resistivity is related not to potential difference V and current i, but to
electric field E and current density J.
E
High resistance
Definition of resistivity
r
J

Units V/m over A/m2 = Vm/A = ohm-meter =W m
Note that the ability for current to flow in a
material depends not only on the material, but
on the electrical connection to it.
1

Note use (re-use) of  for conductivity.
r
NOT surface charge density.

Low resistance
Definition of conductivity
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More on Resistivity

Since resistivity has units of ohm-meter, you might think that you can just
divide by the length of a material to find its resistance in ohms.
R  r / L?
since
E  V / L and J  i / A
resistivity is r 
Rr

L
A
E V /L

 RA / L
J i/ A
Resistance from resistivity
Dependence on temperature: you can imagine that a higher temperature of
a material causes greater thermal agitation, and impedes the orderly flow
of electricity. We consider a temperature coefficient a: r - r0  r0a (T - T0 )
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Resistivity of a Resistor
5.
A.
B.
C.
D.
E.
Three resistors are made of the same
material, with sizes in mm shown below.
Rank them in order of total resistance,
4
greatest first.
I.
I, II, III.
I, III, II.
II, III, I.
II, I, III.
III, II, I.
4
II.
5
Each has
square
cross-section
2
6
III.
3
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Ohm’s Law
R
V
i
Ohm’s law is an assertion that the current through a device is always directly
proportional to the potential difference applied to the device.

A conducting device obeys Ohm’s law when the resistance of the device is
independent of the magnitude and polarity of the applied potential difference.

A conducting material obeys Ohm’s law when the resistivity of the material is
independent of the magnitude and direction of the applied electric field.
4
Does not obey Ohm’s Law
Slope = R
R=1000
Slope
= 1/RW
Potential difference (V)

2
0
-2
-4
-2
0
2
Current (mA)
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Electric Power
dU
dt (watts). Recall also that
 Recall that power is energy per unit time,
for an arrangement of charge, dq, there is an associated potential energy
dU = dqV.
 Thus,
dq
P  V  iV
Rate of electrical energy transfer
dt
P
Units: 1 VA = (1 J/C)(1 C/s) = 1 J/s = 1 W

In a resistor that obeys Ohm’s Law, we can use the relation between R and
i, or R and V, to obtain two equivalent expressions:
P  i2R
P

2
Resistive dissipation
V
R
In this case, the power is dissipated as heat in the resistor.
October 17, 2007
Superconductivity
In normal materials, there is always some resistance, even if low, to current
flow. This seems to make sense—start current flowing in a loop (using a
battery, say), and if you remove the battery the current will eventually slow
and stop.
 Remarkably, at very low temperatures (~4 K) some conductors lose all
resistance. Such materials are said to be superconductors. In such a
material, once you start current flowing, it will continue to flow “forever,”
like some sort of perpetual motion machine.
 Nowadays, “high-temperature” superconductors have been discovered that
work at up to 150 K, which is high enough to be interesting for
technological applications such as giant magnets that take no power,
perhaps for levitating trains and so on.

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Ohm’s Law
6.
The three plots show voltage vs. current (so the
slope is R) for three kinds of device. What are
the devices?
A.
Resistor, superconductor, diode
Diode, superconductor, resistor
Resistor, diode, superconductor
Diode, resistor, superconductor
Superconductor, resistor, diode
C.
D.
E.
II.
Potential difference (V)
B.
I.
III.
Current (mA)
October 17, 2007
How Do Batteries Work?
A battery is a source of charge, but also a source of voltage (potential
difference).
 We earlier saw that there is a relationship between energy, charge, and
voltage U  qV .
 Thus, a battery is a source of energy. We describe a battery’s ability to
create a charge flow (a current) as an electromotive force, or emf.
 We need a symbol for emf, and we will use an E, but it needs to be
distinguishable from electric field, so we will use a script E.

The unit of emf is just the volt (V).
 Other sources of emf are, for example, an electric generator, solar cells, fuel
cells, etc.
Here is a case where two emf
Ea
i
sources are connected in opposing
R
i
directions. The direction of i
Eb
indicates that Ea > Eb. In fact, emf
i
a charges emf b.

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Summary









Current, i, is flow of charge (charge per unit time), units,
amperes (A).
Net current into or out of a junction is zero.
 
i   J  dA
Current density, J, (current per unit area) is a vector.
J is proportional to the density of charge carriers, ne, and 

J

ne
v
d
the drift speed of the carriers through the material.
Resistance, R, (units, ohms, W) is the proportionality
V
R

between voltage V applied, and current, i.
i
Ohm’s Law states that R is a constant. It is not always a
constant, but if not, the device does not obey Ohm’s Law.
1
E
Resistivity (r) and conductivity () are properties of


r
r
J
materials. Resistivity units, ohm-meter.
L
Rr
Resistance is related to resistivity by
A
V2
2
Electric power P (units watts, W) is P  iV . For resistors: P  i R P 
R
October 17, 2007