Electric Circuits (Only Kidding)

February 22, 2019November 8, 2022

201902220351 Short Review III (Resistance)

Definition.

Resistance $R$ is a measure of the opposition of an object to the flow of electric charges. Its physical meaning is the potential difference (p.d.) $V$ to be applied across a conductor in order for a current $I$ to flow through it. Mathematically,

$\displaystyle{R=\frac{V}{I}}$

The unit of resistance is ohm, $\displaystyle{\Omega}$ .

By comparing units,

$[\mathrm{resistance}]=\displaystyle{\frac{[\mathrm{potential\,difference}]}{[\mathrm{current}]}}$ ,

i.e. $1\,\Omega =1\,\mathrm{V\,A^{-1}}$ .

Measuring Resistance

By voltmeter-ammeter method (also by a multimeter). This method must contain experimental error, but we can reduce the errors by using different circuit connections.

voltmeter_ammeter_method

For your information: experimental techniques

In the left diagram, the voltmeter gives a correct reading of p.d. $V$ across the resistor. But the ammeter gives a wrong reading of current $I$ because $I$ includes the additional current passing through the voltmeter. By definition, $\displaystyle{R=\frac{V}{I}}$ , the calculated resistance is smaller than the actual one. But if the resistor has small resistance $R$ , the current passing through the voltmeter is small, and hence the error is reduced.

In the right diagram, the ammeter gives a correct reading of current $I$ passing through the resistor. But the voltmeter gives a wrong reading of p.d. $V$ across the resistor because $V$ includes the additional p.d. across the ammeter. By $\displaystyle{R=\frac{V}{I}}$ , the calculated resistance is larger than the actual one. But if the resistor has large resistance $R$ , the p.d. across the ammeter is small, and hence the error is reduced.

Ohm’s Law

Ohm’s law. The potential difference across the ends of a conductor is directly proportional to the current flowing through it ( $V\propto I$ ), provided that temperature and other physical conditions are unchanged.

Not all conductors obey Ohm’s law, such exception is called non-ohmic. By convention, Ohm’s law qualifies as “a law with exception”, though some persons might even not regard it as a law.

Concept Test

1. Which of the following statements is/are correct?

I. $V=IR$ , where $R$ varies by $V$ or $I$ , is equivalent to Ohm’s law.
II. $\displaystyle{R=\frac{V}{I}}$ defines the resistance of any material, be it ohmic or non-ohmic.
III. Ohm’s law is obeyed when the curve plotted on a $V$ – $I$ graph is a straight line passing through the origin.

A. I and II only
B. I and III only
C. II and III only
D. I, II and III

2. The figures below show the $V$ – $I$ or $I$ – $V$ graphs of copper wire, filament lamp, diode, and dilute sulphuric acid respectively. Comparing_IV_characteristic_of_materials_copper_wire

Which of the following statements is/are true?

I. Both copper wire and filament lamp satisfy Ohm’s law.
II. The diode allows current to flow in only one direction, as long as the potential difference across it does not exceed the breakdown voltage.
III. The current flowing through dilute sulphuric acid is directly proportional to the potential difference applied across it.

3. Which of the following statements is wrong?

A. An electrical conductor has no resistance when there is no current passing through it.
B. The resistance of semiconductors decreases when temperature increases.
C. The resistance of superconductors drops to zero when temperature is extremely low.
D. The output voltage of a battery is lower than its e.m.f. because of its internal resistance.

Answers:

Explanation:

II is correct because of the definition. III is correct because Ohm’s law states that for any ohmic conductor, $V\propto I$ , i.e. $V=\mathrm{const.} \times I$ . This constant is the slope of its $V$ – $I$ graph, and recall the equation $y=mx$ is a straight line passing through the origin. I is wrong because if $R$ varies according to $V$ or $I$ , $V$ is not directly proportional to $I$ .
I is wrong because in the $V$ – $I$ graph of filament lamp the curve is not straight, indicating $V$ is not directly proportional to $I$ . This is because the resistance of filament increases as its temperature increases. Thus, the condition of Ohm’s law, i.e., constant temperature, is not even satisfied. II is correct because at $V>V_{\mathrm{break}}$ the current $I$ is positive, i.e., it is in one direction. III is wrong because the curve is not a straight line passing through the origin, i.e. there is no current even though there is voltage $V$ ( $0<V<V_{\mathrm{back}}$ ) passing through it.
Options B, C, D are correct statements. Output voltage of a battery means the p.d. across an external circuit; while the e.m.f. of a battery is the energy imparted by the source per unit charge passing through it, where some portion of this energy is “lost” in its internal resistance. A is wrong because resistance exists in conductors, whether there is current passing through it or not.

Factors Affecting Resistance

Temperature $T$ :

$T\uparrow\, \Rightarrow R\uparrow$ .

Length $l$ , thickness/cross-sectional area $A$ :

$l\uparrow\,\Rightarrow R\uparrow$ or $A\downarrow\,\Rightarrow R\uparrow$ .

Resistivity $\rho$ : Each material has its own constant resistivity $\rho$ , defined at a certain temperature.

Combining all factors, at constant temperature:

$\displaystyle{R=\rho\frac{l}{A}}$

is the resistance (an extrinsic property, depends on physical dimension of materials).

$\displaystyle{\rho=\frac{RA}{l}}$

is the resistivity (an intrinsic property of materials).

By dimensional analysis,

$\displaystyle{[\mathrm{resistivity}]=\frac{[\mathrm{resistance}][\mathrm{area}]}{[\mathrm{length}]}}=\displaystyle{\frac{\mathrm{\Omega\times m^2}}{\mathrm{m}}}=\mathrm{\Omega \,m}$ .

Concept Test

Which of the following statements concerning resistance of conductors of the same composite material is correct?
1. If the cross-sectional area $A$ of a conductor is constant, its resistance $R$ is inversely proportional to it length $l$ .
2. If the length $l$ of a conductor is fixed, its resistance $R$ is inversely proportional to its cross-sectional area $A$ .
3. For a constant current $I$ flowing through a conductor, resistance $R$ is directly proportional to the potential difference $V$ across it.
4. For a constant potential difference $V$ across a conductor, its resistance $R$ is inversely proportional to the current $I$ passing through it.
A uniform copper wire of length and radius has resistance . What is the resistance of another uniform copper wire of length and radius ?
1. $0.5R$
2. $0.75R$
3. $1.33R$
4. $2R$
Which of the following statements concerning resistivity is correct?
1. Resistivity of a material depends on its temperature.
2. Resistivity of a material depends on its physical dimension.
3. Resistivity of insulators is smaller than that of conductors.
1. I only
2. I and II only
3. I and III only
4. II and III only

Answers:

Explanation:

A is wrong and B is correct because of $\displaystyle{R=\rho\frac{l}{A}}$ . C and D are wrong because $R=V/I$ being the definition of resistance, it indicates that $R$ is constant, independent of applied voltage and current, when the conductor is in constant temperature and of the same composite material.
$R=\displaystyle{\rho \frac{l}{A}}=\rho\frac{l}{\pi r^2}$ . The new resistance $R'=\displaystyle{\rho\frac{3l}{\pi (1.5r)^2}}=\frac{4}{3}\bigg( \rho\frac{l}{\pi r^2}\bigg)=1.33R$ .
I is correct because the higher the temperature, the higher the resistivity. Resistivity is a constant only when defined at a certain temperature. II is incorrect. Resistivity is independent of physical dimension. Do not confuse it with resistance. III is incorrect because insulators’ resistivity is larger than conductors’, as their names suggest. Remark. Resistivity and temperature of a material are intrinsic properties; resistance and physical dimension of a material are extrinsic properties. Intrinsic property affects intrinsic property; extrinsic property affects extrinsic property; but intrinsic property and extrinsic property do not affect each other.

February 22, 2019May 8, 2023

201902220301 Short Review II (Voltage, Electromotive Force, and Potential Difference)

Voltage, p.d. and e.m.f.

Definition.

Voltage $V$ across two points is the change in electric potential energy $U$ per unit charge passing between the points.

The unit of voltage is volt (V). By comparing units, $\displaystyle{[\mathrm{voltage}]=\frac{[\mathrm{potential\,energy}]}{[\mathrm{charge}]}}$ , i.e., $1\mathrm{\,V}=1\mathrm{\,J\,C^{-1}}$

Remark. Equivalently, (i) voltage $V$ across two points is the change in electric potential $V$ between the points. In other words, (ii) voltage $V$ across two points is the potential difference $\Delta V$ (p.d.) between two points.

$V\stackrel{(\mathrm{def})}{=}\displaystyle{\Delta \bigg(\frac{U}{q}\bigg) =\frac{U_1}{q_1}-\frac{U_2}{q_2}\stackrel{(\mathrm{i})}{=}V_1-V_2\stackrel{(\mathrm{ii})}{=}\Delta V\,\mathrm{(p.d.)}}$

Definition.

Electromotive force (e.m.f.) $\varepsilon$ of a power source is the electrical energy per unit charge supplied by the source, when there is charge passing through it.

Remark. The e.m.f. of a source is measured when it is in open circuit, i.e., the source is not in use such that no current is being drawn.

Voltage $V$ refers to e.m.f. $\varepsilon$ when describing a power source, e.g., a cell.

Voltage $V$ refers to p.d. $\Delta V$ when describing an external circuit component, e.g., a load.

Measuring Voltage

By voltmeter, connected in parallel to a component in a circuit. An ideal voltmeter should have infinite resistance.

Voltage in Series and Parallel Circuits

In a series circuit, the sum of the potential differences across each load is equal to the e.m.f. of the power source.

In a parallel circuit, the potential difference across each load is the same as the e.m.f. of the power source.

Cells in Series and Parallel

In a series arrangement, the e.m.f. add up.

In a parallel arrangement, the currents add up.

February 21, 2019October 24, 2022

201902210854 Short Review I (Electric Current)

Definition.

An electric current is a flow of electric charge (through a conductor). We express electric current $I$ by the total electric charge $Q$ flowing through a cross-sectional area per unit time $t$ .

$\displaystyle{I=\frac{Q}{t}}$

One unit current is of one ampere ( $1\,\mathrm{A}$ ).

$\displaystyle{[\mathrm{current}]=\frac{[\mathrm{charge}]}{[\mathrm{time}]}}$ , so $1\,\mathrm{A}=1\,\mathrm{C\,s^{-1}}$ .

Concept Test

An electric current can result from
1. the movement of atoms.
2. the movement of electrons.
3. the simultaneous movement of positive charges and electrons.
  1. I only
  2. III only
  3. I and II only
  4. II and III only
A direct current (d.c.) of flows through a wire. How much charge passes through the wire in 30 minutes? And how many electrons flow through it?
1. $24\,\mathrm{C}$ ; $3.84\times 10^{-18}$ electrons
2. $24\,\mathrm{C}$ ; $1.5\times 10^{20}$ electrons
3. $1440\,\mathrm{C}$ ; $2.304\times 10^{-16}$ electrons
4. $1440\,\mathrm{C}$ ; $9\times 10^{21}$ electrons
Suppose, on average, passengers arrive at Central station from Chai Wan station by MTR every hour. The journey takes up minutes, with average speed . The average train frequency is one every minutes. Which of the following statements is/are correct?
1. The total number of passengers on this journey by MTR at any instant is $1900$ .
2. $3800$ passengers arriving in Central per hour is analogous to a current passing through a cross-sectional area.
3. Average train speed $70\,\mathrm{km/h}$ is analogous to the current in a conducting wire segment.
4. The average train frequency is analogous to the current flowing through a circuit.
1. I and II only
2. I and III only
3. II and III only
4. III and IV only

Answers:

Explanation:

(I) is wrong because atoms are neutral and do not carry charge. Movement of atoms is not a flow of charge, viz. current. (II) is correct because electrons are charge-carriers. (III) is correct. An example is electrolytes with positive ions and negative ions as charge-carriers.
Charges passing through the wire $Q=It=0.8\times 30\times 60=1440\,\mathrm{C}$ (Caution: use SI-unit). An electron $e$ has $1.6\times 10^{-19}\,\mathrm{C}$ of negative charge. So $1\,\mathrm{C}$ of charge consists in $\displaystyle{\frac{1}{1.6\times 10^{-19}}}=6.25\times 10^{18}$ electrons. Thus $1440\,\mathrm{C}$ corresponds to $1440\times 6.25\times 10^{18}=9\times 10^{21}$ electrons. Remark: Options A and C do not make sense, how can an electron be split into pieces?
Draw an analogy like this: (i) trains $\sim$ charge carriers; (ii) passengers $\sim$ charge; and (iii) railway $\sim$ circuit. (2) is correct because current $I=\displaystyle{\frac{Q}{t}}\sim \displaystyle{\frac{3800\mathrm{\,passengers}}{1\,\mathrm{hr}}}$ . (I) is correct because the total number of passengers on board a train at any instant is $3800\times$ time needed for one journey $=3800\times 0.5=1900$ . (III) is wrong because the average train speed $\sim$ the flow of charge carriers $\neq$ the flow of charge. (IV) is wrong because average train frequency $\sim$ density of charge carriers (i.e., electrical conductivity) $\neq$ the current.

Current Direction

By definition, current is the flow of charges (e.g., carried by electrons).

(Charge carriers can be positive, e.g., holes in semi-conductors, positive ions in electrolytes.)

In nature, current is due to the flow of negative electrons from the negative (-ve) terminal to the positive (+ve) terminal of a power source.

An old convention, which is wrong, lasts to date:

Conventional electric current is a flow of positive charge from the +ve terminal to the -ve terminal of a battery.

Conventional current direction is opposite to the direction of electron flow.

External_circuit_direction_of_potential_current_electrons

Measuring Current

By ammeter (also by a current sensor with data-logger, or by galvanometer, which is used for large current, large voltage, and any resistance, like a multimeter), connected in series to a component in a circuit. An ideal ammeter should have zero resistance.

Current in Series and Parallel Circuits

In a series circuit, the current is the same at all points.

In a parallel circuit, the sum of currents passing through each branch is equal to the current in the main circuit.