Electric Circuits 1

• Electric Current

• Resistance and Ohm’s Law

and resistivity

•Energy and Power in Electric

Circuits

• Resistors in Series and

Parallel

•Equivalent resistive circuits

1

INTRODUCTION: Electrical circuits are part of everyday human life.

e.g. Electric toasters, electric kettle, electric stoves

All electrical devices need electric current to operate.

In this section we study charges in motion called electric current.

What is an Electric Current?

Electric current is the rate at which electric charges move

through a wire or a conductor (flow of electric charges).

PARTICLE CHARGE MASS

Electron 1.6 10-19 C 9.11 10-31 kg

Proton 1.6 10-19 C 1.67 10-27 kg

Neutron No charge 1.67 10-27 kg

Structure of the atom:

-Protons and Neutrons fixed in the

Nucleus

-Electrons move in different energy

levels. Valence electrons are free to

move

2

Electronic Current

Conventional Current

Direction of

positive flow

Time interval

dt

t

dtIq

dt

dqI

0

S.I UNITS: Charge in Coulombs (C)

Time in Seconds (s)

Current in Amperes (A)

If a charge, dq, passes through a given cross-section of a

conductor in a time, dt, then a current, , is said to have passed,

where:

3

ELECTROMOTIVE FORCE (EMF)

Any device which causes Charge separation to occur is said to

be a source of electromotive force or EMF.

i.e. A battery with EMF, , provides energy to the charges to make

them move through a wire.

S.I unit: VOLT (V)

A battery that is disconnected from any circuit has an

electric potential difference between its terminals that is

called the electromotive force or EMF:

Remember – despite its name, the EMF is an electric

potential or voltage, not a force.

Electric circuit: A closed path through which charge can

flow, returning to its starting point, is called an electric

circuit. i.e. A closed path is required for current to flow. 4

A battery uses chemical reactions to produce a potential difference

between its terminals. It causes current to flow through the

flashlight bulb similar to the way the person lifting the water causes

the water to flow through the paddle wheel.

When the switch is closed , it provides a closed path for electric

charges to flow.

i.e. the battery sends out positive charges from the positive terminal

and accepts them at the negative terminal.

5

Electric Current

The direction of current flow – from the positive terminal

to the negative one – was decided before it was realized

that electrons are negatively charged. Therefore, current

flows around a circuit in the direction a positive charge

would move; electrons move the other way. However, this

does not matter in most circuits.

Conventional

current

Electronic current

Conventional current

always tries to flow from

the positive terminal to

the negative terminal of

the battery. The symbol

of the battery:

6

We use circuit symbols to make understanding of circuits easier.

Shown are some common circuit symbols:

7

Ammeter

Consider a circuit of battery connected to a light bulb

as shown:

A

Bulb I

If we measure the current with an ammeter we find that

it has same finite value everywhere in the circuit.

e.g. 1 A or 0.01 A

If another bulb is added to the circuit, current will be

different.

i.e. Something about the light bulb limits the size of

current flow in the circuit.

The light bulb has some Impedance 8

IMPEDANCE: What restricts the current flow?

As charges move through a material they experience some

opposition to their flow. The degree of difficulty of current flow is

measured in terms of the Impedance of the material.

If a current I flows through a material

when an EMF, , is applied to the ends of

the material, then the material has an

Impedance, Z, given by: IZ

IZ

or

The S.I. Unit of Impedance is the Ohms

There are three basic circuit components which give rise to Impedance

For a pure resistive circuit: Z = R 9

RESISTANCE (R) Energy is dissipated in a

Resistor in the form of heat

CAPACITANCE (C) Energy is stored in a Capacitor

in the form of an Electric field

INDUCTANCE (L) Energy is stored in an inductor

in the form of a Magnetic field

Resistance and Ohm’s Law

Under normal circumstances, wires present some resistance to the

motion of electrons. Ohm’s law relates the voltage to the current:

Be careful – Ohm’s law is not a universal law and is

only useful for certain materials (which include most

metallic conductors).

Ohm’s Law: The current flowing through

a conductor is directly proportional to

the potential difference across the ends

of a conductor.

IRV

IV

IV

)constant(

IR OR

10

Linear Resistance

I

GradientR

1

εat

volt)at (

1

GradientR

Non-linear Resistance

I

R

IGradient

1

e.g. Thermistors, Diodes,

light bulbs 11

If the graph of current as a

function of applied EMF is Linear,

then the resistance is a constant.

Such conductors are said to obey

Ohm’s Law and are referred as

OHMIC

If the graph of current as a function

of applied EMF is non-linear, then

the resistance varies with the

applied EMF. Such resistors do not

obey Ohm’s Law (NON-OHMIC)

For such resistors, it is common to

define the dynamic resistance at

some desired EMF

Resistance and Ohm’s Law

Solving for the resistance from

Ohm’s Law, we find:

The units of resistance, volts per

ampere, are called Ohms:

Two wires of the same length and diameter will have

different resistances if they are made of different

materials.

This property of a material is called the resistivity, . The

resistance, R, of a wire of length, L, and cross-sectional

area, A, is given by:

IR

OR

12

Resistance and Ohm’s Law

The difference between

insulators, semiconductors,

and conductors can be clearly

seen in their resistivities:

13

Resistance and Ohm’s Law

In general, the resistance of materials goes up as the

temperature goes up, due to thermal effects. This

property can be used in design of thermometers.

Resistivity decreases as the temperature decreases,

but there is a certain class of materials called

superconductors in which the resistivity drops

suddenly to zero at a finite temperature, called the

critical temperature TC. e.g. Mercury 4.2 K,

Highest temperature at which Superconductivity has

been observed is 125 K.

14

Exercise 1: Suppose a charge of 20 C drifts through a conductor of

cross-sectional area, A, in 2.0 s. (i) Calculate the current through the

conductor, (ii) How many electrons will pass through this conductor

in 1.0 s to produce a current of 10A?

Exercise 2:

(i) How much current will flow through a lamp that has a resistance

of 60 when connected across a 12 V supply?

(ii) What is the resistance of an electric frying pan that draws 12 A

when connected to a 120 V circuit?

15

20 30

12

12 V

Exercise 3: For the circuit

shown, find the current

through and potential drop

(voltage) across each

resistor.

Energy and Power in Electric Circuits

Power is defined as the rate at

which the electrical energy is

converted to heat, P = dW/dt

The potential difference, V, between the ends of a wire is defined as

the work (potential energy), dw, required to move a charge , dq, from

one end of the wire to the other. i.e.

VdqdW

q

dW

q

dUV

Potential Difference = Work(Energy)/Charge

Idtdq

dt

dqI

Also, Current:

If this Power is supplied for a time, t, then the amount of Energy

converted to heat:

Energy = Power × Time

tR

VRtIVIttPE

22

S.I. Units: Energy (Joules), J S.I. Units: Power (Watts), W 16

R

VRIVI

dt

dqV

dt

Vdq

dt

dWP

22

Exercise 4: Calculate the power being

dissipated in each resistor in the circuit

Current flowing in the circuit:

A 4.050

20

R

VI

IRV

Power dissipated in the 10 : W6.1104.0 22 RIP

Power dissipated in the 40 : W4.6404.0 22 RIP

Total Power Dissipated: W8 4.66.1 TP 17

When the electric company sends you a bill, your usage is quoted

in kilowatt-hours (kWh). They are charging you for energy use, and

kWh is a measure of energy.

tPE

Resistors in Series and Parallel

18

Resistors connected end to end are said to be in series.

They can be replaced by a single equivalent resistance

without changing the current in the circuit.

In series circuit, same current flows through each resistor

21-4 Resistors in Series and Parallel

Since the current through the series resistors must be

the same in each, and the total potential difference is the

sum of the potential differences across each resistor,

...321 VVVV

We find that the equivalent resistance is:

321

321

RRRR

I

V

I

V

I

V

I

V

eq

19

Resistors in Series and Parallel

Resistors are in parallel when they are across the same potential

difference; they can again be replaced by a single equivalent

resistance:

20

21-4 Resistors in Series and Parallel Using the fact that the potential difference across each resistor

is the same, and the total current is the sum of the currents in

each resistor,

Note that this equation gives you the inverse of the resistance, not

the resistance itself!

...321 IIII

We find:

321

321

321

1111

...

RRRR

R

V

R

V

R

V

R

V

IIII

eq

eq

If just two resistors in parallel, then:

Sum

Product

21

21

RR

RRReq

21

Resistors in Series and Parallel

22

If a circuit is more complex, start with

combinations of resistors that are either purely

in series or in parallel.

Replace these with their equivalent resistances;

as you go on you will be able to replace more

and more of them.

EXAMPLE 1: Find the equivalent resistance between points A and B

Solution:

RCED = 4+ 2 = 6

RCD = 2

2

1

12

6

3

1

6

11

CDR

RAB = 8

23

EXAMPLE 2: Find the equivalent resistance between points A and B

of the circuit shown if each resistor is 2

Solution: To simplify the problem let us label

each resistor a, b, c, d, e, and f.

n.b. The resistors a and b are in series and

the combination is in parallel with resistor c.

The resultant of abc is in series with e.

a b

c

d e

f

x

y

z

u v Rxyz = 2+ 2 = 4

Rxz = 4×2/4+2 = 8/6

Rxzv = 8/6 + 2 = 20/6

Rxuv = 2 + 2 = 4

Rxv = (20/6 × 4)/((20/6)+ 4) = 1.8

24

25

Exercise 5: Consider the circuit shown with

three resistors, R1 = 250.0 , R2 = 150.0

and R3 = 350.0 connected in parallel to a

24.0 V battery. Find:

(i) the current supplied by the battery,

(ii) the current through each resistor.

Exercise 6: An electric heater draws a steady 15.0 A from a 120 V line.

(i) How much power does it require to operate? and

(ii) How much does it cost per month (30 days) if it operates for

3.0 hours per day and the electricity company charges 9.2 cents per

kWh?