Physics 121: Electricity & Magnetism – Lecture 7Current & Resistance

Dale E. GaryWenda Cao

NJIT Physics Department

October 17, 2007

Definition of Current Current is the flow of

electrical charge, i.e. amount of charge per 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 to move will create a current, but a typical way is to attach a battery 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).

Units: ampere 1 A = 1 C/s

Total current in

11 A

Total current out

3 A = 8 A

8 A

October 17, 2007

Current in a CircuitWhat is the current in the wire

marked i in the figure below?

Total current in

11 A

Total current out

3 A = 8 A

8 A

October 17, 2007

Current At JunctionsWhat is the current in all of the wire

sections that are not marked?

8 A

2 A5 A 3 A

6 A

October 17, 2007

Try One Yourself1. What is the current in the wire

section marked i?

A. 1 A.B. 2 A.C. 5 A.D. 7 A.E. Cannot determine from information given.

5 A

2 A 2 A

3 A1 A

6 A

i

October 17, 2007

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 use the symbol . The relationship between current and current density is

J

AdJi

Small current densityin this region

High current densityin this region

October 17, 2007

Drift Speed Let’s look in detail at one happens when we connect a battery to a

wire to start current flowing.

+

_

1.5 Vbattery

Signal travels throughthe wire at the speed of light

Thermal motionsof electrons—no

net drift.Electrons driftin directionopposite to i

Current

October 17, 2007

Drift Speed The drift speed is tiny compared with thermal motions. Thermal motions (random motions) have speed Drift speed in copper is 104 m/s . Let’s relate drift speed to current density.

m/s106thv

A

L

n

dneAvdt

dqAdJi

nALenVeVeV

Nq

Total charge q in volume V

density of charge carriers

vdtvL d

time to drifta distance L

tenAvq d

dvneJ

ne is carrier charge density

+e means J and vd in same directione means J and vd in opposite directions

October 17, 2007

Increasing the Current2. When you increase the current in a

wire, what happens?

A. The number of charge carriers stays the same, and the drift speed increases.

B. The drift speed stays the same, and the number of charge carriers increases.

C. The charge carried by each charge carrier increases.

D. The current density decreases.

October 17, 2007

Resistance Resistance is defined to be . That is, we apply a voltage V, and

ask 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.

i

VR

+

_

1.5 Vbatteryi

i large, soR small

wire

If we apply the voltage to a less conducting material, such as glass, the current will be tiny so R is very large.

i

i small, soR large

glassfilament

The unit of resistance is the ohm, . (Greek letter omega.)

1 ohm = 1 = 1 volt per ampere = 1 V/A

RV CircuitDiagram

Resistor

October 17, 2007

Current Through a Resistor3. What is the current through the

resistor in the following circuit, if V = 20 V and R = 100 ?

A. 20 mA.B. 5 mA.C. 0.2 A.D. 200 A.E. 5 A.

RV CircuitDiagram

October 17, 2007

Current Through a Resistor4. If the current is doubled, what

changes?

A. The voltage across the resistor doubles.B. The resistance of the resistor doubles.C. The voltage in the wire between the battery and

the resistor doubles.D. The voltage across the resistor drops by a factor of

2.E. The resistance of the resistor drops by a factor of

2.

RV CircuitDiagram

October 17, 2007

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 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 and current density J.

J

E

Units V/m over A/m2 = Vm/A = ohm-meter =m

Definition of resistivity

Note that the ability for current to flow in a material depends not only on the material, but on the electrical connection to it.

High resistance

Low resistance

1

Definition of conductivityNote use (re-use) of for conductivity.

NOT surface charge density.

October 17, 2007

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.

?/ LR

AiJLVE / and / since

LRAAi

LV

J

E/

/

/resistivity is

A

LR 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 :

)( 000 TT

October 17, 2007

Resistivity of a Resistor5. Three resistors are made of the

same material, with sizes in mm shown below. Rank them in order of total resistance, greatest first.

A. I, II, III.B. I, III, II.C. II, III, I.D. II, I, III.E. III, II, I.

44

52

63

I.

II.

III.

Each hassquare

cross-section

October 17, 2007

Ohm’s Law 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.

-2 0 2Current (mA)

4

2

0

-2

-4

Pot

enti

al d

iffe

renc

e (V

)

Slope = R

Slope = 1/R

i

VR

R=1000

Does not obey Ohm’s Law

October 17, 2007

Electric Power Recall that power is energy per unit time, (watts). Recall

also that for an arrangement of charge, dq, there is an associated potential energy dU = dqV.

Thus,

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:

In this case, the power is dissipated as heat in the resistor.

dt

dUP

iVVdt

dqP Rate of electrical energy transfer

Units: 1 VA = (1 J/C)(1 C/s) = 1 J/s = 1 W

RiP 2

R

VP

2

Resistive dissipation

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.

October 17, 2007

Ohm’s Law6. The three plots show voltage vs. current

(so the slope is R) for three kinds of device. What are the devices?

A. Resistor, superconductor, diodeB. Diode, superconductor, resistorC. Resistor, diode, superconductorD. Diode, resistor, superconductorE. Superconductor, resistor, diode

I.

II.

III.

Current (mA)

Pot

enti

al d

iffe

renc

e (V

)

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 . 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.

qVU

Here is a case where two emf sources are connected in opposing directions. The

direction of i indicates that Ea > Eb. In fact, emf a charges

emf b.

R

Ea

Eb

i

i

i

October 17, 2007

Summary Current, i, is flow of charge (charge per unit time),

units, amperes (A). Net current into or out of a junction is zero. Current density, J, (current per unit area) is a vector. J is proportional to the density of charge carriers, ne,

and the drift speed of the carriers through the material. Resistance, R, (units, ohms, ) is the proportionality

between voltage V applied, and current, 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.

Resistivity () and conductivity () are properties of materials. Resistivity units, ohm-meter.

Resistance is related to resistivity by Electric power P (units watts, W) is . For

resistors:

AdJi

dvneJ

i

VR

J

E

1

A

LR

iVP RiP 2R

VP

2