XMLECTURE CURRENT OF ELECTRICITY AND DC CIRCUITS

Ver 1.0 © Chua Kah Hean xmphysics 1

XMLECTURE

13 CURRENT OF ELECTRICITY AND DC CIRCUITS NO DEFINITIONS. JUST PHYSICS.

13.1 Voltages .................................................................................................................................... 2

13.1.1 Electromotive Force (emf) .................................................................................................. 2

13.1.2 Potential Difference (pd) ..................................................................................................... 3

13.2 Current ...................................................................................................................................... 3

13.2.1 Conventional Current .......................................................................................................... 4

13.2.1 Drift Velocity ....................................................................................................................... 5

13.3 Resistance ................................................................................................................................ 6

13.3.1 Ohm’s Law ......................................................................................................................... 7

13.3.2 Resistivity ........................................................................................................................... 8

13.4 Power ....................................................................................................................................... 9

13.5 Internal Resistance and Terminal Potential Difference ............................................................ 11

13.5.1 Maximum Efficiency.......................................................................................................... 13

13.5.2 Maximum Power ............................................................................................................... 14

13.6 I-V Characteristic Graph .......................................................................................................... 16

13.6.1 Ohmic Devices ................................................................................................................. 17

13.6.2 Filament ........................................................................................................................... 18

13.6.3 NTC Thermistor ................................................................................................................ 19

13.6.4 Light-Dependent Resistor ................................................................................................. 19

13.6.5 Diode ................................................................................................................................ 20

13.7 DC Circuits .............................................................................................................................. 21

13.7.1 Resistors in Parallel and Series ........................................................................................ 22

13.7.2 Potential Divider Principle ................................................................................................. 23

13.7.3 Slide Wire Potentiometer .................................................................................................. 25

Appendix A Circuit Rules ................................................................................................................ 32

Appendix B Practice Circuits ........................................................................................................... 33

Online resources are provided at https://xmphysics.com/coedc/

https://xmphysics.com/coedc/


13.1 Voltages

An electric current is a flow of charges. An electric current in an electrical circuit is a flow of charges

round and round a loop. To produce such a current, an emf source is needed.

13.1.1 Electromotive Force (emf)

In every emf or voltage source, there is a non-electrical

force Femf acting on some mobile charge carriers1, that

result in excess positive and negative charges residing at

the positive and negative terminals respectively. These

excess charges establish an electric field which is directed

from the positive terminal towards the negative terminal

inside the emf source. Once this field is established, Femf

must do work against the electrical force FE to ferry

charges across the terminals. A positive charge therefore

gains EPE when it is “lifted” from the negative to the

positive terminal (inside the emf source). The EPE gained

per unit charge is called the emf of the voltage source.

The emf of a voltage source is measured in volts2. 1 V

corresponds to 1 J C-1. For example, for a 3 V dry cell, 3 J

of chemical energy is converted into 3 J of electrical

(potential) energy for every C of charge passing through the battery.

1 E.g. in a chemical cell, it is the “chemical force” pushing on ions in the electrolyte. In a generator, it is the magnetic force pushing electrons in the armature windings. 2 Don’t be misled by its name. An emf may be associated with an electric field and forces, but it is defined as a

voltage, not a force.

emf source resistor

E

emf source

Femf

FE Femf

FE


13.1.2 Potential Difference (pd)

If a resistor is connected across an emf source, a current flows

through the resistor. This is because the connecting wires

(assumed to be perfect conductors) causes the same voltage

that is the emf of the source to be across the terminals of the

resistor as well. Notice that the current flows from the positive to

the negative terminal of the resistor. A positive charge therefore

loses EPE when it makes its way from the positive to the

negative terminal of the resistor. The potential difference (pd)

(measured in volts) across the resistor thus represents the EPE

lost per unit charge in the resistor.

For example, if the pd across the filament of a light bulb is 3 V, it

means that 3 J of EPE is converted into 3 J of heat and light

energy, for every C of charge passing through the filament.

13.2 Current

Current is the net flow of electrical charges. Measured in amperes (symbol A), 1 A corresponds to

1 C s−1.

Consider an isolated tungsten filament in a light bulb before it was connected to a battery. As a

conductor, the filament is buzzing with free electrons moving at very high speeds of the order of

106 m s-1. The electrons are always colliding with the massive ions of the material, bouncing off at

Artist’s impression of the path taken by an

electron in the absence of an electric field

resistor

FE

FE

E


random directions. Since everything is random, at the end of the day, as many electrons moved

forward as backward. There is no net flow of electrons. In other words, there is no electric current.

Now let’s connect the filament to a battery so that a constant leftward electric field E is developed

inside the filament. The electrons will now experience a constant electric force, goading them

rightward as they continue to be bounced in all directions. What results is a very slow drift of the

electrons (amidst the random collisions) as a group in the rightward direction. This net flow of electrons,

constitutes an electric current.

The drift speed is typically of the order of 10-4 m s-1. So it takes hours for an electron to trudge its way

from a battery to a light bulb. Yet a light bulb lights up the moment the switch is closed. This is because

the electric field is set up throughout the circuit immediately (in fact, this signal propagates at the

speed of light). The bulb lights up the moment the electrons in the filament starts moving.

13.2.1 Conventional Current

An electric current is a flow of charges. But there are two kinds of charges: positive and negative. It

turns out that a 1 C s-1 net flow of positively charged particles in one direction produces the exact

same electromagnetic effect as a 1 C s-1 net flow of negatively charged particles in the opposite

direction. As such, scientists have chosen to define the direction of current to be the direction of flow

of positive charges. This came to known as the conventional current.

E

Artist’s impression of the path taken by an

electron in the presence of an electric field

conventional current conventional current


In metals, the current is carried by electrons. So while we describe the electrical current to be from

the positive terminal of the battery to the negative, the electrons in the copper wire actually flow from

the negative terminal of the battery to the positive terminal.

It is not uncommon to have currents formed by moving positive charges. For example, in an ionized

gas (plasma), or in an electrolyte, the current is carried by positive and/or negative ions. In a

semiconductor (e.g. silicon, germanium), the current is carried by valence holes (+ve) and/or

conduction electrons (−ve).

13.2.1 Drift Velocity

Consider a cylindrical wire in which electrons drifting at speed v produce a current I. What is the

relationship between v and I?

In time duration t, each free electron in the wire would have drifted forward a distance of vt. This

means that the charge carriers in the shaded volume would have passed through cross sectional area

A of the wire.

This cylindrical space has a volume of

( )( )V A v t

If there are n electrons per unit volume, the number of electrons in this volume is

N nAv t

Since each electron carries a charge of 191.60 10 Ce , the amount of charge in that volume is

Q nAv t e .

vt

A v

I


Since current is charge per unit time, the current is

I Q t

nAv te t

nAve

In general, the current could be carried by charged particles other than electrons. So

I nAvq

where q is the charge of each mobile charge carrier.

One insight we can gain from this formula is this. For the same current, the drift velocity is higher for

an insulator than a conductor, since an insulator has much lower mobile charge concentration than a

conductor. This means a stronger electric field and thus potential difference is required to produce

the same current. It is harder to push a current through an insulator because the fewer charge carriers

have to be pushed harder. This is the primary reason for the higher resistance in insulators.

13.3 Resistance

An electric current in a tungsten filament results from the drift of electrons. We can understand the

electrons’ drift as a result of the acceleration caused by the electric field in the filament. From the

energy perspective, we can say that electrons are losing EPE in exchange for KE.

As they drift through the filament, however, the electrons collide with the massive ions of the tungsten

metal, and lose their newly gained KE to the vibrational energy of the ions. The tungsten becomes

hot and eventually glows. This is how the EPE of the electrons is ultimately converted into heat and

higher

EPE

lower

EPE

Artist’s impression of electrons passing

their KE to the metal ions


light in the filament. Anyway, the way electrons keep losing their energy represents a resistance to

the flow of electric current. The filament is an electrical resistor.

13.3.1 Ohm’s Law

So to maintain a current I through a resistor, a pd across the resistor V is required. It is known that for

some materials, especially metals, at a given temperature, V is proportional to I.

V IR

This relationship is called Ohm’s “Law”.

Ohm’s Law is one of those laws that are not necessarily obeyed all the time (similar to Hooke’s Law

and the Ideal Gas equation). But whether Ohm’s Law is obeyed or not, the concept of resistance is

now defined. The resistance between the points in a circuit is simply the ratio of the pd (between the

two points) to the current (through the two points).

Thus, given any mysterious two-terminal component, one can determine its resistance by applying a

known voltage V across it, and measuring the resulting current I through it. The resistance is simply

the V to I ratio.

VR

I

A word of caution: In the previous chapter, the symbol V was used to denote electric potential, and

V for difference in potential between two positions or change in potential. In this chapter, the same

symbol V is often used to denote both a voltage or potential difference, and electric potential at a point.

Electricians are not very cautious with their nomenclature to say the least.

R

higher

potential

I

lower

potential

V

I


13.3.2 Resistivity

Consider a cylindrical carbon wire of length L and cross-sectional area A. When a pd of V is

maintained across its two ends, a current I passes through the wire. The resistance of the wire is

VR

I .

What if L is doubled? To maintain the same current as before, the pd must be doubled so that the

electric field in the resistor is maintained (V

Ex

). So the new resistance is 2

' 2V

R RI

. So we

conclude R L .

What if A is doubled? For the same V and electric field strength, the drift velocity of the electrons is

unchanged. However, having double the volume means that there are now twice as many charges

doing the drift. So the new resistance is 1

'2 2

VR R

I . So we conclude

1R

A .

Note that extending L is like connecting resistors in series. So the resistance goes up. Whereas

broadening A is like connecting resistors in parallel. The resistance goes down.

Anyway, since L

RA

, the resistance of a conductor can be written as

LR

A

The constant of proportionality is called the resistivity. It is a property of the material. For example,

copper has a resistivity of 81.7 10 Ωm (at 20°C) compared to 53.5 10 Ωm for carbon. So

obviously copper wires have much lower resistance than carbon wires of the same dimensions.

Resistivity of a material is related to the mobile carrier concentration and the atomic structure of the

material. Resistivity is affected by temperature.

In a nutshell, the resistance of a component depends both on the material it is made of and its

dimensions. A short, fat, cold copper wire is a better conductor than a long, skinny, hot tungsten wire.

L

A I

V


Material Category Resistivity at 20°C / Ω.m

Copper Conductor 81.7 10

Carbon (graphite) Conductor 53.5 10

Water Conductor 2 to 200

Silicon Semiconductor 26.4 10

Skin (dry) Insulator 43 10

Glass Insulator 1010 to 1014

Tabulated above are the resistivity of some common materials. By the way, it is the large disparity in

resistivity between conductors and insulators that makes it is so easy to confine electric currents in

electrical circuits.

13.4 Power

From the energy perspective, the emf source converts energy from non-electrical to electrical form.

In the external circuit, the resistors convert energy from electrical to non-electrical form. For example,

in a dry-cell, chemical forces “pump” the electrons to higher EPE. In the tungsten filament, light and

heat is produced when the electrons lose their EPE (during collisions with the metal ions).

Since current is rate of flow of charges Q

It

and pd across the filament is EPE converted per unit charge W

VQ

It follows that the rate of energy conversion in the filament is IV

higher

potential

lower

potential

electrical to

non-electrical

non-electrical

to electrical

I I

V

http://www.xmphysics.com/13.4


From Ohm’s Law, the formula for power dissipation can be expressed in three forms.

2

2

P VI

P I R

VP

R

watch video at xmphysics.com



13.5 Internal Resistance and Terminal Potential Difference

An external load of resistance R is powered by a battery of emf E.

Since R is connected directly across the battery, the potential difference across the resistor is always

be equal to E, regardless of the resistance value of the resistor. But a zero resistance battery is

mythical beast, like the frictionless plane and the massless pulley.

In practice, all emf sources carry some inherent internal resistance. For example, a chemical cell must

push ions through the electrolyte. An electric generator must push the current through the coil

windings with thousands of turns. A solar cell must push electrons and holes through the silicon

substrate. If the resistance of the electrical path in the emf source is substantial, then a significant

potential difference is required to push the current through the emf source itself. In casual speak, we

say that part of the emf of the battery E is used up by the pd across the internal resistance r, lowering

the terminal potential difference Vt that is ultimately available to the external resistance R. This

sentence is encapsulated by the equation below

tV E Ir

R

E I

R

r E I



Plotted as a graph, we get a downward sloping straight line whose steepness corresponds to the

internal resistance r. So the terminal pd is equal to the emf only if zero current is drawn from the

battery. This happens if the circuit is left open. So the emf is also called the open circuit terminal pd.

The larger the current drawn (when the external resistance is decreased), the lower the terminal pd.

Another expression for Vt is as follows:

t

RV E

R r

This expression comes from the potential divider principle since the emf is divided between the

internal and external resistances. It is thus clear that the terminal pd is substantially lower than the

emf of the battery when the internal resistance is comparable to the external resistance.


Vt

E

I

Ir



13.5.1 Maximum Efficiency

Suppose we are now stuck with a battery with an internal resistance r. We are interested in how the

efficiency of our circuit is affected by the external resistance R.

The efficiency of the circuit can be written as

2

2 2

power dissipated in

power dissipated in and

R I R R

R r R rI R I r

So clearly, the larger the external resistance, the higher the efficiency. As the internal resistance

becomes a smaller fraction of the total resistance, the power wasted in the internal resistance also

becomes a smaller fraction of the total power. Note that when R r , the efficiency is exactly 50%!

100%

efficiency,

external resistance, R

50%

r

http://www.xmphysics.com/13.5.1


13.5.2 Maximum Power

But what about the output power Pout? By Pout, we mean the power dissipated in the external

resistance. So

( )( )

out tP V I

R EE

R r R r

Perhaps you feel that having a large external resistance R would result in a large Pout. If R is too large,

Vt approaches open circuit voltage of E but I approaches zero. Pout approaches zero!

Perhaps you changed your mind. We should go for the smallest external resistance instead. But if the

external resistance is too small, I approaches short circuit current of E

r but Vt approaches zero. Pout

approaches zero again!

So what is the value of R that maximizes Pout? Let return to the math.

2

2( )out

RP E

R r

By differentiating the equation and solving for 0dP

dR or otherwise, it can be shown that maximum

Pout is achieved when the external resistance R matches the internal resistance r.

2

max 2

2

( )

4

rP E

r r

E

r

Pout

external resistance, R r

E2/4R



In theory, if your battery has zero internal resistance, infinite output power is attained by an external

resistance of zero. In practice, however, maximum power of 2

4

E

r is attained by an external resistance

that matches the internal resistance of the battery. This is called the maximum power theorem.

Recall that when R r , the efficiency is only 50% (because half the power is wasted in the internal

resistance). That’s the price to pay if high power output is what you’re after. For example, if you want

your headphone to be playing at the loudest possible volume, the electrical resistance of your

headphone should be designed to match the internal resistance of the circuitry driving your

headphone.


13.6 I-V Characteristic Graph

Suppose you are given this mysterious two-terminal component or device. To investigate its behavior,

you can apply different voltages across its terminals, and measure the resulting currents passing

through it. If you plot the data as a current-voltage graph, you have produced the I-V characteristic

curve for this component.

Suppose the I-V curve for this (fictitious) component looks like this. Can you tell from the curve the

resistance of this component?

Remember that resistance is a simple voltage to current ratio. So the calculation of the resistance of

the component at different operating points is straight forward.

3.06.0 Ω

0.5AR

6.01.6 Ω

3.8BR

9.01.9 Ω

4.7CR

So, clearly this component’s resistance varies from one operating point to another. The resistance at

each operating point is actually equal to the reciprocal of the gradient of the line joining the origin and

that operating point. I like to call these the wiper lines (drawn in blue in the diagram).

The lower the wiper line leans towards the V-axis, the larger the resistance. The higher the wiper line

leans towards the I-axis, the smaller the resistance. Using this idea, even without calculation, we

could have figured out that B C AR R R .

I/A

V/V A (3.0, 0.5)

B (6.0, 3.8) C (9.0, 4.7)



13.6.1 Ohmic Devices

The resistance of a coil of copper wire is constant as long as its dimensions and temperature are kept

constant. Likewise, a carbon film resistor presents a fixed resistance when operated at low power.

Resistors which obey Ohm’s Law (V IR ) are said to be ohmic. Obviously, the I-V graphs for ohmic

resistors are straight lines passing through the origin. So the lines themselves are the “wiper lines”.

The lower the line, the higher the resistance. So 1 2 3R R R .

Most off-the-shelve resistors can be assumed to be ohmic as long as they are operated at low power.

When operated at high power, their temperature may change significantly and their resistances may

be affected.

I/A

V/V

R1

R2

R3


13.6.2 Filament

The filament of an incandescent bulb is typically made of tungsten. Like most metals, the resistivity of

tungsten is highly dependent on its temperature. At higher temperature, the metal lattice vibrates more

vigorously and saps more energy from the passing electrons. It corresponds to a higher resistivity,

resulting in an increase in the resistance of the filament.

As a result, the I-V graph of a filament is a flattening curve at high V values. From the fact that the

“wiper lines” lean more and more towards the V-axis, we can tell that the resistance of the filament

increases with V. This is because as the voltage across the filament increases, so does the current,

and thus the power dissipation ( )P VI . Higher power dissipation results in a hotter filament, which

has a higher resistance. This is why the I-V graph of a filament is straight only at low voltage and

power. Once the voltage and power is high enough and the filament starts to get hot, the graph starts

to flatten.


I/A

V/V

A

B C

RA<RB<RC




13.6.3 NTC Thermistor

A thermistor is a piece of semiconductor. A semiconductor

has very interesting properties. At very low temperatures, a

semiconductor is basically an insulator because it does not

have any mobile charge carriers to carry a current. At higher

temperature, electrons are “shaken loose” from the atoms

and become mobile. So the higher the temperature, the

higher the charge carrier concentrations (more charge

carriers per unit volume). Hence the resistivity of a

semiconductor decreases as temperature rises. Hence the

resistance of a thermistor decreases with temperature.

As the I-V graph reveals, a thermistor has a different constant resistance at different temperatures.

The higher the temperature, the lower the resistance.

13.6.4 Light-Dependent Resistor

An LDR is a piece of semiconductor designed to be exposed

to external illumination. At very low temperatures, a

semiconductor is basically an insulator because it does not

have any mobile charge carriers to carry a current. The

illumination provides the energy to “loosen” some electrons

from the atoms. These electrons are mobile and capable of

carrying a current. So the more brightly the LDR is

illuminated, the higher the charge carrier concentrations

(more charge carriers per unit volume). Hence the resistivity

of the semiconductor decreases with illumination. Hence the

resistance of a LDR decreases with illumination.

As the I-V graph reveals, a LDR has a different constant resistance at different intensity L of the

illumination. The more light it receives, the lower the resistance.

I/A

V/V

T1

T2

T3 T1<T

2<T

3

I/A

V/V

L1

L2

L3 L1<L

2<L

3



13.6.5 Diode

A diode aka p-n junction is another interesting semiconductor device. It is a two-terminal device: one

terminal is called the p junction and the other terminal is called the n junction. When you connect a

diode to an electrical circuit, you better know which terminal is which because the diode allows the

current to flow in one direction only.

If the voltage applied is such that the p junction is at a higher potential than n junction, the diode is

said to be in forward bias. Conversely, if the voltage applied is such that the n junction is at a higher

potential than the p junction, the diode is said to be in reverse bias.

An ideal diode should present zero

resistance in forward bias, and infinite

resistance in reverse bias. Basically, an

ideal diode is a perfect one-way valve.

A practical diode requires a small amount of

forward bias (called the turn-on voltage)

before it switches on. Nevertheless, with

sufficient forward bias, it resistance

decreases rapidly and soon behaves like a

0- wire connection. A practical diode also

allows a very tiny amount of current (called

the reverse current) in reverse bias.

Nevertheless, the resistance remains large

and it behaves practically like an open circuit.

p junction n junction

forward

bias

+ − I

reverse

bias

+ − I=0

short

circuit

open

circuit

V/V

I/A

V/V

I/A


13.7 DC Circuits

Electrical and electronics engineering is basically about getting charges to do work and perform tasks

for us. From power generation to motors to antenna to computers and handphones, electrical circuits

are necessary to channel the charges and current into paths we want them to take. In the H2 syllabus,

we are really only scratching the surface, playing with just batteries and resistors, with an occasional

diode or thermistor. Really basic stuff.


13.7.1 Resistors in Parallel and Series

When resistors are connected in series, both resistors have the same current passing through them.

And the pd across each resistor sum up to be the total pd. So

1 2

1 2

1 2

total

eff

eff

V V V

IR IR IR

R R R

When resistors are connect in parallel, both resistors have the same pd across their terminals. The

currents through each resistor sum up to be the total current. So

1 2

1 2

1 2

1 1 1

total

eff

eff

I I I

V V V

R R R

R R R

The derivation can be easily extended to more than two resistors.

Connecting N resistors (with resistances R1, R2, … RN) in series results in an effective resistance of

1 2 ...eff NR R R R

Connecting N resistors (with resistances R1, R2, … RN) in parallel results in an effective resistance of

1 2

1 1 1 1...

eff NR R R R

R1 R2

V1

I

V2

Vtotal

Reff I

Vtotal

R1

R2

Itotal

V

I1

I2

Reff Itotal

s

V



13.7.2 Potential Divider Principle

Two resistors R1 and R2 are connected in series. If the total voltage across them is Vtotal, how is this

voltage divided between R1 and R2?

Since the current following through them is the same, 1 1

2 2

1 2( )total

V IR

V IR

V I R R

From here, it is obvious that the resistor with the larger resistance grabs a bigger share. In fact, the

voltage ratios follow the resistance ratios. This is called the potential divider principle.

1 1 11

1 2 1 2

2 2 22

1 2 1 2

total

total

total

total

V IR RV V

V IR IR R R

V IR RV V

V IR IR R R

A potential divider is a very frequent occurrence in practical circuits. In fact, a potential divider

(sometimes called a potentiometer) is a standard off-the-shelve three-terminal component, with the

symbol

As the symbol suggests, the third terminal is connected to a sliding contact, called a wiper, moving

over the resistive element.

R1 R2

V1 V2

Vtotal

I

12 V

Vout



For example, in the circuit above, by adjusting the sliding contact, Vout can be varied between 0 V and

12 V. Vout can then be used as a signal to control the brightness of a lamp, or the volume of the stereo.

In the circuit above, a thermistor and a fixed resistor form a potential divider.

thout in

th

RV V

R R

Since the resistance of the thermistor increases with temperature, this circuit can be used to “sense”

the temperature. For example, we can mount this thermistor on your laptop CPU so that the thermistor

can track the temperature of the CPU. This works because Rth (and thus Vout) decreases when the

CPU is hot and increases when the CPU is cool. So Vout could be used as an input signal to a

microcontroller to decide when to turn off the cooling fan.

In the circuit above, a LDR and a fixed resistor form a potential divider.

out in

ldr

RV V

R R

Since the resistance of the LDR increases with illumination, this circuit can be used to “sense” the

intensity of the ambient lighting. For example, we can mount this LDR on the roof top. So Vout would

increase when it is bright as Rldr decreases, and increases when it is dim as Rldr increase. So Vout

could be used as an input signal to a microcontroller to decide when to turn off the lights.

R

Rth

o

o

Vout

Vin

R

Rldr

o

o

Vout

Vin


13.7.3 Slide Wire Potentiometer

A special application of the potential divider is the slide wire potentiometer circuit. It is basically an

elaborate “ruler” for measuring voltages. Since we have voltmeters and DMMs (digital multimeters)

nowadays, I don’t think anybody uses such a circuit anymore. Nevertheless it serves as a decent

introduction to the concept of potential balancing.

The easiest way to understand how this “voltmeter” works is to use an example.

In this example circuit, we have cell of known emf 1 6.0 VE connected across a long thin wire of

uniform cross sectional area, length 60 cmL and total resistance 6.0 ΩR . This is called the

driver circuit of the potentiometer. This cell is often called the driver cell. The wire is called the slide

wire.

Do you notice the other cell of emf 2 3.0 VE ? One end of the cell is connected to the positive end

of the slide wire. The other end is connected to a galvanometer and a jockey. The jockey can be made

to contact any point along the slide wire. This is called the secondary circuit. In practice, E2 is the

unknown emf we are trying to measure. I am revealing its value to you to make the explanation easier.

Do you realize that we are looking at a two-loop circuit driven by two emf sources? In general, solving

such a circuit involves writing two Kirchoff’s Law equations. But you don’t have to worry about that.

Thanks to null deflection!

galvanometer

A B

@ 30 cm

X

E2=3.0 V

E1=6.0 V

C D

jockey

60 cm, 6.0



Consider point X to be the midpoint of AB (the 30 cm mark). Before the jockey is connected, there is

no current flowing between the driver and secondary circuit. So the current in the driver circuit (and

slide wire) can be calculated easily to be 1 3 1.0 AI I . And 3.0 VAXV . On the secondary side, we

have 3.0 VCDV . So before the jockey is connected, we already have AX CDV V . Therefore when the

jockey is connected, no current will flow between the two circuits. And the galvanometer shows null

deflection, as shown below.

A B

@ 30 cm

X

E2=3.0 V

E1=6.0 V

C D

60 cm, 6.0

I1=1.0 A

I2=0.0 A

I3=1.0 A

VAX

=VCD

=3.0 V

A B

@ 30 cm

X

E2=3.0 V

E1=6.0 V

C D

60 cm, 6.0

I1=1.0 A

I2=0.0 A

I3=1.0 A

VAX

=VCD

=3.0 V


Now consider point Y to be at the 20 cm mark of the wire. Before the jockey is connected, the current

in the slide wire can be calculated easily to be 1 3 1.0 AI I . And 2.0 VAYV . On the secondary

side, we have 3.0 VCDV . So before the jockey is connected, we have AY CDV V . When the jockey

is subsequently connected, a current must flow from C to A to increase the current in the slide wire

so that VAY can be increased to match VCD. And the galvanometer shows a deflection. In case you’re

interested, I have presented the solution to this circuit below. You don’t have to know how I obtained

these answers. But you can check and verify that this is indeed the correct solution because (1) the

currents tally up at each junction and (2) the voltages tally up in each loop.

A B

@ 20 cm

Y

E2=3.0 V

E1=6.0 V

C D

60 cm, 6.0

I1=1.0 A

I2=0.0 A

I3=1.0 A

VAY=2.0 V, VCD=3.0 V

A B

@ 20 cm

Y

E2=3.0 V

E1=6.0 V

C D

60 cm, 6.0

I1=0.75 A

I2=0.75 A

I3=1.5 A I1=0.75 A

VAY=VCD=3.0 V


Now consider point Z to be at the 40 cm mark of the wire. Before the jockey is connected, the current

in the slide wire can be calculated easily to be 1 3 1.0 AI I . And 4.0 VAZV . On the secondary

side, we have 3.0 VCDV . So before the jockey is connected, we have AZ CDV V . When the jockey

is connected, a current must flow from A to C to decrease the current in the slide wire so that VAZ

decreases to match VCD. And the galvanometer shows a deflection in the other direction. In case

you’re interested, I have presented the solution to this circuit below. You don’t have to know how I

obtained these answers. But you can check and verify that this is indeed the correct solution because

(1) the currents tally up at each junction and (2) the voltages tally up in each loop.

A B

@ 40 cm

Z

E2=3.0 V

E1=6.0 V

C D

60 cm, 6.0

I1=1.0 A

I2=0.0 A

I3=1.0 A

VAZ=4.0 V, VCD=3.0 V

A B

@ 40 cm

Z

E2=3.0 V

E1=6.0 V

C D

60 cm, 6.0

I1=1.5 A

I2=0.75 A

I3=0.75 A I1=1.5 A

VAZ=VCD=3.0 V


So do you realize how the potentiometer works now? Basically, to measure E2, we slide the jockey

along the slide wire until we find the null deflection point. Do realize that AX CDV V all the time

regardless of where the jockey makes contact (since A is connected to C and X is connected to D).

What’s special about the null deflection point is that when there is no current flowing between the

driver and secondary circuits (zero current in DX means zero current in AC also), the two-loop circuits

are no longer intertwined with each other. This allows us to analyze the two circuits separately as if

they are not connected. So based on the ratio of the balance length LAX to the total length LAB of the

slide wire, we have

AXAX AB

AB

LV V

L

For this simple example, we also have

2AX CDV V E

1ABV E .

So the emf of the secondary cell is simply 2 1

AX

AB

LE E

L .

Primary circuit: 1

AXAX

AB

LV E

L

Secondary circuit: 2CDV E

The connection: AX CDV V

A B

@ LAX

X

E2

E1

C D

LAB, R

I1

I2=0.0 A

I3=I1 I1

VAX=VCD


In practice, the circuit can be slightly more complicated. For example, a protective resistor R2 may be

added to the secondary circuit. Without this resistor, a large current may be flowing in the secondary

circuit as we are still searching for the null deflection (by tapping the jockey too close to A). However,

having R2 does not affect our calculation of E2 at all. This is because at null deflection, there is no

current flowing through R2 so the pd across R2 is zero. So VCD is equal to E2 at null deflection, whether

R2 is there or not.

We also often include a resistor R1 in the driver circuit. This would make VAB a fraction of E1. Usually

R1 is added so that the balance length LAX is increased. This is to reduce the percentage uncertainty

in the measurement of LAX and ultimately of E2.

With R1, E2 is obtained through the following equations:

Primary circuit: AXAX AB

AB

LV V

L and

1

1

AB

RV E

R R

Secondary circuit: 2CDV E


A B

@ LAX

X

E2

E1

C D

LAB, R

I1

I2=0.0 A

I3=I1 I1

VAX=VCD

R1

R2


The circuit shown above is one where the potentiometer is used to measure the terminal pd of E2. It

looks complicated. But the null deflection saves the day by allowing us to analyze the driver and

secondary circuits separately.

Primary circuit: AXAX AB

AB

LV V

L and

1

1

AB

RV E

R R

Secondary circuit: 32 2 3

3

CD

RV E E I r

R r


A B

@ LAX

X

E2

E1

C D

LAB, R

I1

I2=0.0 A

I3=I1 I1

VAX=VCD

R1

r

R3

I3


Appendix A Circuit Rules

Some circuits are actually quite fun. As long as you are clear of the “rules”, and as long as you have

enough practice.

(1)

The pd across a zero-ohm wire must be zero. Because the tiniest voltage would result in an infinitely

large current. 0

V VI

R . Conclusion: any points connected by a wire are always at the same

electric potential.

Points of the same potential are like points on the same floor in a building.

(2)

A resistor (with non zero resistance) must have a pd across it when a current runs through it. The

current enters the resistor from the higher potential terminal, and exits at the lower potential terminal.

On the other hand, if no current is running through the resistor, then its pd must be zero, and both

ends of the resistor are at the same potential. .0 0V IR I

Resistors are like staircases which may connect a higher floor to a lower floor.

(3)

An emf source is like an elevator that brings charges from ground floor to top floor. Resistors are like

staircases that charges take to make their way back to ground floor. This is why the total emf must

be equal to the total pd in the circuit.

(4)

For a current to flow, a mobile charge must move into the space vacated by another mobile charge.

If the mobile charges at any point in the circuit do not budge, then the flow of charges grinds to a

complete half throughout the circuit. Basically everyone must move together, if not, nobody can move

at all. This is why there is zero current if the circuit is open. This is also why the total current entering

any node is equal to the total current leaving.

(4)

Two branches are parallel if they start at the (potential) point, and end at the same (potential) point.

The pd of the branches are the same.


Parallel branches are like two staircases that start from the same higher floor, and end at the same

lower floor.

Appendix B Practice Circuits

1)

What is the resistance between P and Q?

2)

Evaluate the resistance between i) BC and ii) AC.

P Q R R R

20

50

30

20

50

B

A

C

D

http://www.xmphysics.com/13.B


3)

2016 P1 Q29

A battery of emf 24 V and negligible internal resistance is connected to a network of resistors.

What is the potential difference between junctions X and Y?

4)

What do the voltmeters read if bulb B is fused?

5)

What is the current in the 2 resistor?

20 V

X

6.0 Ω

4.0 Ω

4.0 Ω 8.0 Ω 8.0 Ω

Y

4.0 Ω

9 V

A

B

4

4

2

12 V

12 V


6)

Above are two circuits for determining the unknown resistance R. An ideal ammeter has zero

resistance. An ideal voltmeter has infinite resistance. For most resistance R, both circuits are equally

good. However, if resistance of R is so small that it is comparable to the ammeter’s resistance, then

only one circuit is good. Conversely, if R is so large that it is comparable to the voltmeter’s resistance,

then the other circuit is good. Which is which?

7)

For both circuits, determine the range of voltage across the fixed 10 resistor.

R

E

I

I

R

E

I

I

(X) (Y)

0 to 100

10

9.0 V

10

9.0 V 100


8)

Evaluate the resistance between i) BC and ii) AD.

Answers:

1) R/3

2) 5.77 Ω, 13.7 Ω

3) 4.0 V

4) 0 V across A, 9 V across B

5) 0 A

6) Use X for very small R, Y for very large R

7) 0.82 V to 9.0 V, 0 V to 9.0 V

8) R/2, R

R

B

A

C

D

R

R R

R

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XMLECTURE CURRENT OF ELECTRICITY AND DC CIRCUITS