Day: February 22, 2019

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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_wireComparing_IV_characteristic_of_materials_filament_lamp Comparing_IV_characteristic_of_materials_diode Comparing_IV_characteristic_of_materials_dilute_sulphuric_acid

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:

  1. C
  2. B
  3. A

Explanation:

  1. 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.
  2. 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.
  3. 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

  1. 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.
  2. A uniform copper wire of length l and radius r has resistance R. What is the resistance of another uniform copper wire of length 3l and radius 1.5r?
    1. 0.5R
    2. 0.75R
    3. 1.33R
    4. 2R
  3. 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:

  1. B
  2. C
  3. A

Explanation:

  1. 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.
  2. 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.
  3. 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.
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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.