1Week 1

Electric Current and Voltage

Electric Current

Consider a medium with cross-

section of A m2 with charges moving with

a velocity v from left to right, as pictured in

Figure 1.1. If in a period of time t, qcoulombs cross A in the indicated

direction, we define the average current I

generated by the charge flow to be

Definition of Current: Current is charge in motion.

t

qI

Notes

1. The physical dimension of current is coulomb per second (C/s).

2. The SI unit for current is the ampere (A).

3. The direction of the current I is the same as the direction of the

charge motion

cross section A

velocity v

Figure 1.1

current I

Specifically, if only positive

charges are continuously crossing the

cross-section A, then the resulting

current is solely due to the flow of

these positive charges. If in a period of

time t, q+ coulombs cross A in the indicated direction, we define the

average current due to the flow of

positive charges as

Current due to flow of positive charges

Figure 1.2

t

qI

cross section A

velocity v

current I+

Note

1. The direction of the current I+ is the same as that of the positive

charge motion.

If only negative charges are

continuously crossing the cross-section

A, then the resulting current is solely

due to the flow of these negative

charges. If in a period of time t, q-

coulombs cross A in the indicated

direction, we define the average

current due the flow of negative

charges as

Current due to flow of negative charges

Figure 1.3

t

qI

cross section

A

velocity v

current I-

Note

1. The direction of the current I- is opposite to that of the negative charge

motion.

If both positive and negative

charges are continuously crossing the

cross section A in opposite directions,

then the resulting total current I (the

current measured by an external

ammeter) is given by

Current due to flow of both positive and negative charges

Figure 1.4

Note

1. The direction of the current total current I is taken to be the same as

that of the current I+ (conventional current flow).

III

cross section

A

current I+

current I-

total current I

On the other hand, if both positive

and negative charges are continuously

crossing the cross section A in the

same direction, then the total average

current flow I is given by

Current due to flow of both positive and negative charges

Figure 1.5

Note

1. The direction of the current total current I is taken to be the same as

that of the current I+.

III

cross section

A

current I+

current I-

total current I

In practice, current is measured

using an ammeter. Direct currents are

measured using dc ammeters. A direct

current (abbreviated as dc current) is a

current that flows in one direction only.

The magnitude of a direct current can

be constant or fluctuating but the sign

is unchanged over time.

Current is measured by connecting

an ammeter in the path of the current

flow.

Measuring Current

Figure 1.6

current I

DC Ammeter

A dc ammeter is a polarised

measuring instrument in that it has both

a positive and a negative terminal for

connection to a circuit. The reading

displayed by an ammeter will depend

on how it is connected in the circuit. A

dc ammeter might display a positive

reading when connected one way, but

will display a negative reading when

the meter connections are reversed, or

vice-versa.

Measuring Current (continued)

Figure 1.7

current I

DC Ammeter

current I

DC Ammeter

In the conventional view of current

flow, the positive reading displayed by an

ammeter measuring a current i is

attributed to the flow of positive charges

through the ammeter from the positive

terminal to the negative terminal. A

negative reading displayed by the

ammeter, on the other hand, is attributed

to positive charges flowing through the

ammeter from the negative terminal to

the positive terminal.

Conventional view of current flow

Figure 1.8

current i

DC Ammeter

current i

DC Ammeter

(a) A positive ammeter reading is

attributed to positive charges flowing

from the positive terminal to the negative

terminal of the ammeter.

(b) A negative ammeter reading is

attributed to positive charges flowing

from the negative terminal to the positive

terminal of the ammeter.

In the modern view of current flow,

the positive reading displayed by an

ammeter measuring a current i is

attributed to the flow of electrons through

the ammeter, entering from the negative

terminal and exiting via the positive

terminal. A negative reading displayed by

the ammeter, on the other hand, is

attributed to electrons flowing through

the ammeter from the negative terminal

to the positive terminal.

Modern view of current flow

Figure 1.9

current i

DC Ammeter

(a) A positive ammeter reading is

attributed to electrons flowing from the

negative terminal to the positive terminal.

current i

DC Ammeter

(b) A negative ammeter reading is

attributed to electrons flowing from the

positive terminal to the negative terminal.

While it has now been conclusively

proven that the current flowing in the

metallic conductors of an ammeter is

due to the motion of electrons, to avoid

the confusion of dealing with negative

signs, we will use the conventional view

of current flow when analysing circuits.

Which one to use: the conventional

or the modern view of current flow ?

Figure 1.10

current I

DC Ammeter

current I

DC Ammeter

(a) In the conventional view of current

flow, a positive ammeter reading is

attributed to positive charges flowing

from the positive terminal to the

negative terminal of the ammeter.

(b) A negative ammeter reading is

attributed to positive charges flowing

from the negative terminal to the positive

terminal of the ammeter.

In circuit analysis, we use arrows to

represent ammeters measuring the

currents of interest. The direction of the

arrow points from the point where the (+)

terminal of the ammeter is connected

toward the point where the (-) terminal of

the meter is connected. The direction

pointed to by the arrow is defined as the

currents reference direction.

Reference direction for current

Figure 1.11

(a) Physical circuit showing how the

ammeter is connected to measure the

current flowing through the circuit element.

Ammeter

Circuit

element

i

Circuit

element

(b) The arrow in the schematic circuit represents

the location and relative connection of the

physical ammeter used to measure the

current flowing through the circuit element.

The reference direction of a current

follows the ammeter connection. If the

ammeter connection is reversed, the

direction of the arrow must also be

reversed.

The direction of the arrow does not

change with the current, even though the

current might be reversing its flow with

time.

Reference direction for current (continued)

Figure 1.12

(a) The ammeter connection is now reversed.

Ammeter

Circuit

element

i

Circuit

element

(b) When the ammeter connection is reversed,

then the direction of the arrow in the

schematic circuit follows.

We can connect the ammeter in any

direction we choose. There is no right or

wrong direction for the meter connection.

While the old analogue technology only

measures the positive current, the new

digital ammeters can measure both the

positive and negative currents

Reference direction for current (continued)

Figure 1.13

Positive ammeter reading only means

the meter has been connected in a

way that measures the positive

current, and vice versa.

Ammeter 2

Circuit

element

Ammeter 1

(a) Two ammeters are used to measure the

same circuit current.

Circuit

element

i1 i2

(b) The two ammeters give the same

magnitude reading but their signs are

opposite to one another.

i1 = - i2

Example

Figure 1.14

For each of the hypothetical

conductors shown in Figure 1.14,

determine the magnitude and sign of

the ammeter reading. Assume for

questions (c) to (e) that one positive

or negative charge is equal to 0.5 C.

(b)

1023 electrons/min

DC Ammeter

(c)

10 + charges/sec

DC Ammeter

(a)

1023 electrons/min

DC Ammeter

(e)

6 + charges/sec

DC Ammeter

(d)

6 + charges/sec

DC Ammeter

12 - charges/sec

12 - charges/sec

Solution (Figure 1.14a)

Figure 1.14(a)

The ammeter gives the average

current I

1023 electrons/min

DC Ammeter

t

ne

t

qII

60

1010609.1

2323

mA 27I

Therefore,

I

Solution (Figure 1.14b)

Figure 1.14(b)

The ammeter gives the average

current I

t

ne

t

qII

60

1010609.1

2323

mA 27I

Therefore,

(b)

1023 electrons/min

DC Ammeter

Solution (Figure 1.14c)

Figure 1.14(c)

The ammeter gives the average

current I

t

ne

t

qII

105.0

A 5I

Therefore,

10 + charges/sec

DC Ammeter

Solution (Figure 1.14d)

Figure 1.14(d)

The ammeter gives the average

current I

A 9I

Therefore,

6 + charges/sec

DC Ammeter

12 - charges/sec

t

q

t

qIII

125.065.0

Solution (Figure 1.14e)

Figure 1.14(e)

The ammeter gives the average

current I

A 3I

Therefore,

t

q

t

qIII

125.065.0

(e)

6 + charges/sec

DC Ammeter

12 - charges/sec

Instantaneous current

Figure 1.15

If the time t gets smaller and smaller,

then, in the limit t goes to zero, the ratio

q/t approaches the slope of the curve

at point t; that is,

idt

dq

t

q

t

0lim

i is called the instantaneous current and

is time-dependent. To explicitly show the

time dependence, we sometimes write

dt

dqti )(

time

q(t)

q

t

Slope = dq/dt

t

Example

Figure 1.16

Find and plot i(t) if q(t) is given by the graph in Figure 1.16.

t, s1 2 3 4 5 6 7 8

1

2

3

4

5

- 1

- 2

- 3

- 4

- 5

q(t)

Solution

Figure 1.17

t, s1 2 3 4 5 6 7 8

1

2

3

4

5

- 1

- 2

- 3

- 4

- 5

q(t)

For 0 t 2 s, slope of line is

202

041

t

qm

024

442

t

qm

745

433

t

qm

024

444

t

qm

C/s = 2A

C/s = 0 A

C/s = - 7 A

C/s = 0 A

For 2 t 4 s, slope of line is

For 4 t 5 s, slope of line is

For 5 t 6 s, slope of line is

For 6 t 8 s, slope of line is

C/s = 1.5 A5.168

)3(05

t

qm

Solution (continued)

Figure 1.18

t

q

dt

dqi

s

s

s

s

s

t

t

t

t

t

A

A

A

A

A

ti

87

75

54

42

20

5.1

0

7

0

2

)(

For the case where the charge q(t) varies linearly with time, we can write

Hence, the current i(t) is given by the piecewise function

t, s1 2 3 4 5 6 7 8

1

2

3

4

5

- 1

- 2

- 3

- 4

- 5

i(t)

Relationship between current and charge

We can determine the charge that passes through the cross-

section A of the medium in Figure 1.19 in the time interval - to t if we integrate current with respect to time; that is

cross section A

Figure 1.19

current i

t

ditq )()(

where i is current in amperes

q is charge in coulombs

t is time in seconds

Relationship between current and charge

By breaking the integration into two time segments, namely,

from to 0, and from 0 to t, we can write Eq.() as

cross section A

Figure 1.19

current i

tt

qdiditq0

)0()()()(

where

0

)()0( diq

Example

The current flowing through a circuit element is given as 50 mA. Compute

the amount of charge transferred in 100 ns.

Solution

To solve this problem using calculus, we assume that the given current is

the instantaneous current i(t) that has a constant value of 50 mA for times

t 0 and a zero value for times t < 0. Mathematically, we write

0

0

mA 50

0)(

t

tti

The graph for i(t) is shown in Figure 1.20.

i(t)

50 mA

t

Figure 1.20

Solution (continued)

i(t)

50 mA

t

Figure 1.20

ns

dinsq

100

)()100(

ns

didi

100

0

0

)()(

The amount of charge transferred between times t = 0 and t = 100 ns is

ns

dd

100

0

3

0

10500

ns100031050 nC 5

Example

The current in a wire is given by

otherwise

s 6t0

A

0

2sin10

)(

t

ti

Find the total charge passing a cross section of the wire from t = 0 and t = 6 s.

i(t)

Figure 1.21

Solution

A plot of the current i(t) is shown Figure 1.22.

i(t) A

t (s)

10

0 2 4 6

Figure 1.22

)di()0()6(

6

0

qq

)di()0(

0

-

q

The total charge up to time t = 6 is given by

the expression

where

)d2

10sin()6(

6

0

q

C 12.73

Figure 1.23

Since i(t) = 0 for t < 0, we have

0 d0)0(

0

-

q

12

cos10

6

0

Hence

A plot of q(t) is shown in Figure 1.23

0 1 2 3 4 5 6

2.12

4.24

6.37

8.49

10.61

12.7312.732

0

q t( )

60 t

Charge transferred by a constant current flow

T

0

0

I0 dd

IT

If the current flow is constant, that is if the current i(t) = I for times

between t = 0 and t = T, where I and T are both constants, then

the charge transferred up to time T is given by

T

diTq )()(

If we write q(T) Q, then we obtain

ITQ

ITQ

Example

The current flowing through a circuit element is given as 50 mA. Compute

the amount of charge transferred in 100 ns.

Solution

Since the current is constant we can simply obtain Q using the relationship

Hence

nC 5 ns 100mA x 50 IT Q

Definition

Potential Difference and Voltage

The work w done by the electrical system in moving a charge q from

a point A to another point B is determined by the potentiaI difference

(or simply, voltage) that exists between A and B. Quantitatively, the

potential difference between A and B (indicated by the voltage vAB) is

defined to be

q

wqvAB

moved charge ofamount

B A to from charge movingin donework

The unit for potential difference is energy/charge, joules per coulomb in

the MKS system, but to honour Count Alessendro Volta, we use the

special name volt (V) for this unit. Thus, we say that the potential

difference between point A and point B is 1 volt if 1 joule of work is

done in moving a unit charge (+1 C) from A to B.

I =q/t

vAB

A

B

Figure 1.24

Measuring voltage

For convenience we simplify the circuit

drawing as shown in Figure 1.25b. The

symbol v represents the voltmeter reading,

and the (+) and (-) signs associated with the

voltage v correspond to the location of the

(+) and (-) terminals of the voltmeter in the

measurement. The location of the +/- signs

defines the reference direction for the

voltage.

In practice, we measure voltage with a

voltmeter (VM), as shown in Figure 1.25a.

The meter will indicate the potential

difference between A and B.

VoltmeterCircuit element

A

B

(a)

v

Circuit

element

A

B

(b)

Figure 1.25

+/- Notation for voltage

In Figure 1.25(b) we have

expressed the potential difference

across the circuit element by

marking both ends of the circuit

element with polarity symbols: a + at one end and a at the other end. This is called the +/- notation

for voltage labelling. With this

notation, the convention is that the

+ represents the first subscript and the the second subscript of the voltage. (a) (b)Figure 1.26

C

v2

B

v1

A

#1

#2

v1

B

A

#1

#2

C

v2

Figure 1.26b shows the voltage across the two circuit elements marked in

this manner. To differentiate between the two voltages we can label them use

either a numerical or an alphabetic subscript.

Arrow notation for voltage

It is sometimes convenient to use arrows

to define voltage reference directions.

Then, as in Figure 1.27, the head of the

arrow is the point of measurement and

the tail is the point of reference. With this

notation, the convention is that the

arrowhead represents the location of the (+) terminal of the measuring voltmeter

and the tail represents the (-) terminal.Figure 1.27

+ 15 V

-10 V 5 V

v1

A B C D

-15 V

v2

E

#1 #2 #3

By using the arrow notation, we can treat

the voltages as vectors; hence the rules

for vector addition and vector subtraction

can be applied for the voltages.

0 V

Arrow notation for voltage (continued)

Example

Referring to Figure 1.27, find vA, vB, vC,

v1, and v2.

Answer

vA = + 15 V ; vB = - 10 V

vC = + 5 V ; vD = - 15 V

v1= vB vA

= (-10 V) (15 V)

= - 25 V

Figure 1.27

+ 15 V

-10 V 5 V

v1

A B C D

-15 V

v2

E

#1 #2 #3

0 V

v2 = vC vD= ( 5 V) (- 15 V) = 20 V

Double subscript notation

We often use double subscripts to indicate the

voltmeter connections. The first subscript indicates

where the (+) terminal is connected in the circuit,

and the second subscript indicates where the (-)

terminal is connected. vAB

vCB

B

A

#1

#2

CC

B

VM1

A

#1

#2

Figure 1.28

(a) (b)

VM1

Using the double-subscript notation, the

voltages measured by voltmeters VM1 and

VM2 in Figure 1.28(a) would be written as vABand vCB, respectively. Thus, according to the

double-subscript notation, vAB in Figure

1.28(b) is the potential difference measured

between points A and B, and vCB is the

potential difference measured between points

C and B.

Rule on changing the order of voltage subscripts

vAB = + 10 V

A

B

vBA = - 10 V

Changing the order of the subscripts changes the sign of the voltage

Example

What is the voltage vBA for the circuit shown in Figure 1.29?

Answer

Figure 1.29

Rule on intermediate points for voltage subscripts

Voltage at intermediate points in a circuit when measured in the same

direction adds algebraically.

Thus, in the circuit shown in Figure 1.30 we can write, for example, the voltage

vAE as follows:

vAE = vAB + vBC + vCD + vDE

Figure 1.30

A CB#1 #2 #3 #4

D Eor

vAE = vAC + vCD + vDE

vAE = vAB + vBD + vDE

or

Rule on intermediate points for voltage subscripts (continued)

Example

Figure 1.31

Find the voltage vAC in Figure 1.31

C

B

A

#1

#2

vAB = 10 V

vCB = -5 V

Rule on intermediate points for voltage subscripts (continued)

Answer

Figure 1.31

From the figure, we can write

vAC = vAB + vBC

= vAB + (-vCB)

Hence, C

B

A

#1

#2

vAB = 10 V

vCB = -5 V

= vAB - vCB

vAC = (10 V) (- 5 V) = 15 V

Exercise

Refer to Figure 1.32. If vAB = + 2 V and vCB = - 1 V, find vBA and vCA.

Rule on intermediate points for voltage subscripts (continued)

C

B

A

#1

#2

Figure 1.32

Answer

vBA = - vAB = - 2 V

vCA= vCB+ vBA = (- 1 V) + (-2 V) = - 3 V

Rule on intermediate points for voltage subscripts (continued)

C

B

A

#1

#2

Figure 1.32

Double subscript notation (continued)

Figure 1.33

Note

Avoid using arrows to show

voltage reference direction

when using the double-

subscript notation.

vAB

vCB

B

A

#1

#2

C

(a)

vAB

vCB

B

A

#1

#2

C

(b)

Question

Which of the two figures on

the right shows the correct

application of the arrow

notation for voltage reference

direction, based on the double

subscripts written for the

voltages?