Current and Resistance

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Define the current.

Understand the microscopic description of current.

Discuss the rat at which the power transfer to a device in an electric current.

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2-1 Electric current

2-2 Resistance and Ohm’s Law

2-3 Current density, conductivity and

resistivity

2-4 Electrical Energy and Power

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Whenever electric charges of like signs move, an electric current is said to exist.

The current is the rate at which the charge flows through this surface◦ Look at the charges flowing perpendicularly to a surface of

area A

The SI unit of current is Ampere (A) 1 A = 1 C/s

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QI

t

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∆Q is the amount of charge that passes through this area in a time interval ∆ t,

the average current Iav is equal to the charge that passes through A per unit time

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We define the instantaneous current I as the differential limit of average current:

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The direction of the current is the direction positive charge would flow◦ This is known as conventional current direction

In a common conductor, such as copper, the current is due to the motion of the negatively charged electrons

It is common to refer to a moving charge as a mobile charge carrier . A charge carrier can be positive or negative. For example, the mobile charge carriers in a metal are electrons.

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Charged particles move through a conductor of cross-sectional area A

n is the number of charge carriers per unit volume

n A Δx is the total number of charge carriers

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The total charge is the number of carriers times

the charge per carrier, q

◦ ΔQ = (n A Δx) q

The drift speed, vd, is the speed at which the

carriers move

◦vd = Δx/ Δt

Rewritten: ΔQ = (n A vd Δt) q

Finally, current, I = ΔQ/Δt = nqvdA

OR

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the average current in theconductor

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If the conductor is isolated, the electrons

undergo random motion

When an electric field is set up in the

conductor, it creates an electric force on the

electrons and hence a current

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The zig-zag black line represents the motion of charge carrier in a conductorThe net drift speed is small

The sharp changes in direction are due to collisions

The net motion of electrons is opposite the direction of the electric field

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Consider a conductor of cross-sectional area A carrying a current I. The current density J in the conductor is defined as the current per unit area. Because the current I = nqvdA,

the current density is:

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the current density is proportional to the electric field:

Where σ the constant of proportionality & is called the conductivity of the conductor.

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If the field is assumed to be uniform, the potential difference is related to the field through the relationship

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express the magnitude of the current density in the wire as

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Where ,J = I/A, we can write the potential difference as

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The quantity R = ℓ/σA is called the resistance of the conductor. We can define the resistance as

the ratio of the potential difference across a conductor to the current in the conductor:

,

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In a conductor, the voltage applied across

the ends of the conductor is proportional to

the current through the conductor

The constant of proportionality is the

resistance of the conductor

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VR

I

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Units of resistance are ohms (Ω)

◦1 Ω = 1 V / A

Resistance in a circuit arises due to collisions

between the electrons carrying the current with

the fixed atoms inside the conductor

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Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents

This statement has become known as Ohm’s Law◦ ΔV = I R

Ohm’s Law is an empirical relationship that is valid only for certain materials◦Materials that obey Ohm’s Law are said to be

Ohmic

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An ohmic device The resistance is

constant over a wide range of voltages

The relationship between current and voltage is linear

The slope is related to the resistance

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Non-Ohmic materials are those whose resistance changes with voltage or current

The current-voltage relationship is nonlinear

A diode is a common example of a non-Ohmic device

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The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A

◦ ρ is the constant of proportionality and is called the resistivity of the material

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LR

A

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For most metals, resistivity increases with increasing temperature

◦With a higher temperature, the metal’s constituent atoms vibrate with increasing amplitude

◦The electrons find it more difficult to pass through the atoms

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For most metals, resistivity increases approximately linearly with temperature over a limited temperature range

◦ ρ is the resistivity at some temperature T◦ ρo is the resistivity at some reference

temperature To To is usually taken to be 20° C is the temperature coefficient of

resistivity

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)]TT(1[ oo

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Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance

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)]TT(1[RR oo

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o o(T) 1 T T

o

1= temperature coefficient of resistivity

T

o oR(T) R 1 T T

RA

A class of materials and compounds whose resistances fall to virtually zero below a certain temperature, TC

◦ TC is called the critical temperature

The graph is the same as a normal metal above TC, but suddenly drops to zero at TC

In a circuit, as a charge moves through the battery, the electrical potential energy of the system is increased by ΔQΔV

As the charge moves through a resistor, it loses this potential energy during collisions with atoms in the resistor◦The temperature of the resistor will

increase

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Consider the circuit shown

Imagine a quantity of positive charge, Q, moving around the circuit from point A back to point A

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Point A is the reference point◦ It is grounded and its potential

is taken to be zero

As the charge moves through the battery from A to B, the potential energy of the system increases by QV◦ The chemical energy of the

battery decreases by the same amount

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As the charge moves through the resistor, from C to D, it loses energy in collisions with the atoms of the resistor

The energy is transferred to internal energy

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When the charge returns to A, the net result is that some chemical energy of the battery has been delivered to the resistor and caused its temperature to rise

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The rate at which the energy is lost is the power

From Ohm’s Law, alternate forms of power are

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QV I V

t

22 V

I RR

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The SI unit of power is Watt (W)◦I must be in Amperes, R in ohms and V

in Volts

The unit of energy used by electric companies is the kilowatt-hour◦This is defined in terms of the unit of

power and the amount of time it is supplied

◦1 kWh = 3.60 x 106 J

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The same potential difference is applied to the two

lightbulbs shown in Figure .Which one of the following statements is true?

(a) The 30-W bulb carries the greater current and has the higher resistance.

(b) The 30-W bulb carries the greater current, but the 60-W bulb has the higher resistance.

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(c) The 30-W bulb has the higher resistance, but the 60-W bulb carries the greater current.

(d) The 60-W bulb carries the greater current and has the higher resistance.

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(c). Because the potential difference ∆V is the

same across the two bulbs and because the

power delivered to a conductor is P= I V, the 60-

W bulb, with its higher power rating, must carry

the greater current.

The 30-W bulb has the higher resistance because

it draws less current at the same potential

difference.

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1. The electric current I in a conductor is defined as

where dQ is the charge that passes through a cross section of the conductor in a time interval dt. The SI unit of current is the ampere (A), where 1 A = 1 C/s.

The average current in a conductor is related to the motion of the charge carriers through the relationship

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where n is the density of charge carriers, q is the charge on each carrier, vd is the drift speed, and A is the cross-

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2. The current density in an ohmic conductor is proportional to the electric field according to the expression

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The proportionality constant σ is called the conductivity of the

material of which the conductor is made. The inverse of & is

known as resistivity ρ (that is, ρ = 1/ σ).

The last equation is known as Ohm’s law, and a material is said

to obey this law if the ratio of its current density J to its applied

electric field E is a constant that is independent of

the applied field.

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3. The resistance R of a conductor is defined as

where ∆V is the potential difference across it, and I is the

current it carries. The SI unit of resistance is volts per

ampere, which is defined to be 1 ohm; that is, 1Ω = 1 V/A.

If the resistance is independent of the applied potential

difference, the conductor obeys Ohm’s law.

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4. If a potential difference ∆V is maintained across a circuit element, the power, or rate at which energy is supplied to the element, is

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Because the potential difference across a resistor is given by ∆V = IR, we can express the power delivered to a resistor in the form

The energy delivered to a resistor by electrical transmission appears in the form of internal energy in the resistor.

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1. The charge carriers in metals areA. electrons.B. positrons.C. protons.D. a mix of protons and electrons.

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2. A battery is connected to a resistor. Increasing the resistance of the resistor

will

A. increase the current in the circuit. B. decrease the current in the circuit. C. not affect the current in the circuit.

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3. A battery is connected to a resistor. As charge flows, the chemical energy of the battery is dissipated as

A. current. B. voltage. C. charge. D. thermal energy

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