3'3Current Electricity

1
3.3 Current Electricity
3.3.1 General Flow Equation
3.3.2 Electromotive Force and Potential
Difference
3.3.3 Resistance and Resistivity
3.3.4 Potential Divider
2
3.3.1 General Flow Equation
This chapter deals with electric charges
. Formal definition of various macroscopic
parameters are defined.

• Electric current I
• An electric current consists of a drift of
  .
• The conventional direction of an electric current
  is that in which there is a net displacement of
  charge.
• An elecrtic current I through a cross-section
  area A is defined as the of charge Q
  passing through it, i.e.,
• Eq.(1)
• The unit of I is .

3

• Charge carriers in solids
• In conductors, currents are constituted by ,
  which are not attracted to any particular atom.
• In semi-conductor, only a small proportion of
  or are free to move.
• Conduction in solid conductors
• In the absence of an electric field,
  electrons in solid conductors can move rapidly
  due to their (or internal ) energy.
• The average speed of the free electrons is very
  , with a typical order of m s-1.
• However, the electronic motion is generally
  due to their frequent collisions with the
  lattice of .
• There is, therefore, no flow of charge and so
  no .

4

• If a battery is connected across the ends of the
  conductor, an is created in the conductor
  which causes the electrons to and hence
  gain energy.
• Due to the collisions with the in the cyrstal,
  however, the electrons and losses
  their energy, whilst the ions gain energy.
  
• The net effect is to transfer energy from the
  battery, via the , to the energy of the
  ions lattice. That is why the temperature of the
  conductor upon conduction of current.
• On account of the collisions, the overall
  acceleration of the electron is .
• The mean speed due to the net motion of the
  electrons is called the .

5

• Drift velocity vd and electric current I
• The following figure shows the volume occupied
  by the free electrons in a conducting element in
  a very short time ?t.
• Number of free electrons passing through A in ?t
  
• Amount of charges ?Q passing through A in ?t
• Current I (amount of charges passing through A
  per unit time)
• Therefore, the drift velocity vd, which is a
  microscopic quantity can be expressed in terms of
  I as
• Eq.(2)

6

• Estimating the drift velocity vd
• Given A 1.1 ? 10-7 m2, n 1029 m-3, e 1.6 ?
  10-7 C, for a current of 1 A,
• vd
• ?
• and the estimated time for an electron to travel
  through 1 m is about .
• In view of such a low speed, why can a light be
  turned on almost instantaneously when a switch is
  closed?

7
Examination questions

• 1997-IIA-21
• Which of the following quantities can be
  increased by increasing the voltage across a
  metal wire?
•  
• (1) the average drift velocity of the conduction
  electrons
• (2) the number of conduction electrons per unit
  volume
• (3) the speed of the electrical signal
•   A. (1) only
• B. (3) only
• C. (1) and (2) only
• D. (2) and (3) only
• E. (1), (2) and (3)

8

• 1999-IIA-23
• A resistance wire is connected across the
  terminals of a battery. Which of the following
  statements is incorrect?
•   A. Before connection to the battery, the
  conduction electrons in the wire move randomly.
• B. After connection to the battery, an electric
  field is set up along the length of the wire.
• C. The conduction electrons in the wire are
  accelerated momentarily in the opposite
  direction to that of the electric field.
• D. The conduction electrons collide with one
  another, giving out heat energy.
• E. The current in the wire is proportional to
  the average drift velocity of electrons.

9

• 2005-IIA-33
• A p.d. of 10 V is applied across a uniform wire
  of resistance 5.0 ?. The number of conduction
  electrons per unit volume of the wire is 1.0 ?
  1028 m-3 and its cross-sectional area is 1.0 mm2.
  What is the average drift velocity of conduction
  electrons in the wire?
• (Given charge on electron 1.6 ? 10-19 C)
• A. 3.1 ? 10-4 m s-1 B. 1.3 ? 10-3 m s-1
• C. 3.1 ? 10-3 m s-1 D. 6.3 ? 10-3 m s-1

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3.3.2 Electromotive Force and Potential
Difference

• Electromotive force ?
• When a current flows out of a battery, positive
  charge moves from potential to potential
  and thus the energy decreases. When the
  charge carriers return to the battery, they have
  used up all of their energy.
• Then, they have to be driven by the battery
  internally from potential to potential.
  The battery is said to have on the charge
  carriers.
• The electromotive force (e.m.f.) ? of a source
  is defined as the imparted by the
  per unit passing through it, i.e.,
• ?
• The unit of ? is .

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• Potential difference V
• The potential difference V between two points in
  a circuit is the amount of changed to
  other form of energy per unit passing
  between the points, i.e.,
• V
• Power supply and power dissipation
• The power P supplied by a source of e.m.f. ? when
  a current I is drawn from it is
• That is,
• Eq.(3)
• Similarly, the power P dissipated across a device
  with potential difference V with a current I
  passes through it is

12
Examination questions

• 1999-IIA-20
• Each of the following three students states a
  conclusion after learning the formulae P IV, P
  I2R, P V2/R in electricity. Which of these
  conclusions is/are correct?
•   (1) The power dissipated by any electric
  appliance can be calculated from P IV.
• (2) The power dissipated by a kettle is
  inversely proportional to its resistance as P
  V2/R.
• (3) The power consumed by a running motor cannot
  be calculated from P I2R.
• A. (1) only
• B. (3) only
• C. (1) and (2) only
• D. (2) and (3) only
• E. (1), (2) and (3)

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3.3.3 Resistance and Resistivity

• Resistance R
• The resistance R of a conductor is defined as
  the of the potential difference V across it to
  the current I flowing through it.
• R Eq.(4)
• The unit of R is .
• The resistance of a device represents its to
  the passage of current.
• The conductance G of a conductor is
  of its resistance, and is measured in .
• Variation of I with V in various electrical
  components
• When a variable p.d. V is applied across an
  electrical component, the current I can be
  measured and the relation between V and I is
  called the of the component.

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• The following shows some examples of I-V graphs
  for metal, electrolytes, semiconductor diode and
  thermistor.
• Ohms law
• Conductors with I-V graphs being straight lines
  through the origin are called or
  conductors, examples of which are .
• Conductors with resistance with applied p.d.
  at constant temperature are said to obey ohms
  law.
• The power P dissipated in an ohmic device is
• P Eq.(5)

15

• Types of resistors
• Fixed resistors have resistance values shown by
  colour markings.
• Variable resistors have the resistance in
  steps or continuously.
• In particular, a slide-wire type variable
  resistor, sometimes called , is shown below.

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• Resistivity
• The resistance of a conductor depends on its
  as well as on the of which it is made.
  To make fair comparisons of the conducting
  abilities of different , quantity independent
  of the of the specimen therefore must be
  considered.
• Experiment shows that the resistance R of a wire
  of length l with constant cross-sectional area A
  behaves like
• or
• Eq.(6)
• where is called the resistivity of the
  material of the conductor.
• Knowing the resistivity of a material the
  of any specimen made of that material can be
  found.

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Examination questions

• 1997-IIA-25
• Two cylinder metal rods, X and Y, are made from
  the same material and have the same mass. The
  length of X is three times that of Y. If
  currents of 1 A and 2 A pass through X and Y
  respectively, the ratio of the power dissipation
  in X to that in Y is
•   A. 91 B. 14 C. 92 D. 34 E. 94

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• 1998-IIA-19
• The variation of current with the voltage
• applied across a device is as shown. What
• is the change in resistance of the device
• when the voltage increases from 1.0 V to
• 2.0 V?
•   A. It decreases by 0.25 W
• B. It increases by 0.25 W
• C. It decreases by 1.50 W
• D. It increases by 0.60 W
• E. It decreases by 0.60 W

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• Effects of temperature on resistance
• Metal
• A rise in temperature in a metal
  the average separation of the ions in the
  metal, which causes a local of the
  crystal lattice. As a result, the free electrons
  collide frequently with the lattice, which
  causes the drift velocity to and hence
  the resistance (resistivity) to .
• Semiconductor
• In semiconductors, the in drift velocity upon
  increased temperature is more than compensated
  when vibration of the atoms causes ,
  releasing charge carriers and thereby produces
  a marked of resistance with temperature rise.

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• Resistor networks
• Resistors in series
• Votage across the resistors V
• The equivalent resistance is
• Req
• Resistors in parallel
• Curret through the resistors I
• The inverse of equivalent resistance is

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• Kirchhoffs laws
• Kirchhoffs first law
• At a junction in a circuit, the current arriving
  the current leaving, i.e., by
  assigning currents arriving as positive, we have
• Kirchhoffs second law
• Round any closed loop in a circuit, the
  algebraic sum of the is equal to the algebraic
  sum of the of current I and resistance R,
  i.e.,

22
Examination questions

• 1990-IIA-33
• In the above circuit, the battery has negligible
  internal resistance. The current I is
•   A. 2/3 A. B. 1 A. C.  4/3 A. D. 2 A.
  E. 6 A.
• 1990-IIA-34
• Which of the following circuits is best used for
  the measurement of a low resistance R?
  (Polarities of meter terminals are given in the
  diagram)
• A. B. C.
• D. E.

23

• 1991-IIA-32
• In the circuit shown, filament lamp L1 and L2
  are identical and are seen to be marked 6 V 18
  W. Which of the following statements is/are
  correct?
•   (1) When switch S is closed, lamp L1 and L2
  glow with the same brightness.
• (2) When switch S is closed, lamp L2 glows with
  its normal brightness.
• (3) When switch S is opened, lamp L1 glows with
  its normal brightness.
•   A. (1), (2) and (3) B. (1) and (2)
  only C. (2) and (3) only
• D. (1) only E. (3) only

24

• 1992-IIA-29
• In the circuit shown, the galvanometer G reading
  is zero. So X has a resistance of
•   A. 15 ? B. 25 ? C. 50 ?
  D. 100 ? E. 150 ?
• 1995-IIA-23
• Two cells of negligible internal resistance are
  connected with two resistors as shown. What is
  the potential difference between X and Y?
•   A. 1.33 V B.  1.67 V C. 2.00 V D. 
  2.33 V E. 2.67 V

25

• 1995-IIA-30
• The machines in a factory consume 10 kW
  electrical power at a voltage of 500 V. If the
  generator is connected to the factory through
  cables of total resistance 0.2 ?, the e.m.f.
  produced by the generator should be
•   A. 500 V B. 501 V C. 502 V D. 504 V
  E. 508 V
• 1996-IIA-24
• In the above circuit, the galvanometer reading
  is zero. If both cells have negligible internal
  resistance, the e.m.f. of the cell E is
•   A. 6 V B. 7 V C. 8 V D. 9 V
  E. 10 V

26

• 1996-IIA-25
• Six wires, each of resistance 1 ?, are joined to
  form a regular tetrahedron ABCD. A current of 2
  A flows across the tetrahedron at A and B.
• Which of the following statements is/are
  correct?
• (1) Points C and D are of the same potential.
• (2) The current in wire AC is 0.5 A.
• (3) The potential drop across AB is 1 V.
•   A. (1) only B. (3) only C. (1) and
  (2) only
• D. (2) and (3) only E. (1), (2) and
  (3)

27

• 1998-IIA-18
• In the above circuit, light bulbs L1, L2 and L3
  consume power at the rate 113. If the
  resistance of L3 is R, the resistance of L1 is
• A. 3R/16 B. R/4 C. 3R/4 D. 4R/9 E. 3R
• 1998-IIA-21
• In the above circuit each cell has negligible
• internal resistance. The voltmeter has a
• resistance of 5 k?. The reading on the
• voltmeter is
•   A. 0 V B. 1.0 V C. 1.2 V
• D. 1.5 V E. 2.0 V

28

• 1999-IIA-16
• The above circuit can be used to find the
  resistance of the resistor R. which of the
  following statements is/are correct?
•  
• (1) The current passing through the ammeter is
  in fact larger than that passing through R.
• (2) The ratio of voltmeter reading to ammeter
  reading is in fact smaller than the resistance
  of R.
• (3) The circuit is suitable for measuring high
  resistance.
•  
• A. (1) only B. (3) only C.
  (1) and (2) only
• D. (2) and (3) only E. (1), (2) and (3)

29

• 1999-IIA-17
• Three resistors of resistance R1, R2 and R3 are
  connected in parallel. It is known that R1gt R2gt
  R3. The equivalent resistance of this
  combination is R. which of the following
  statements is/are correct?
•   (1) Energy dissipated moving 1 C of charge
  through the resistor of resistance R1 is greater
  than that through R3.
• (2) R is smaller than R2.
• (3) If the resistor with resistance R1 is
  removed, the resulting equivalent resistance is
  increased.
•   A. (1) only B. (3) only C. (1) and
  (2) only
• D. (2) and (3) only E. (1), (2) and (3)

30

• 1999-IIA-17
• In the above circuit, the battery has constant
  e.m.f. and negligible internal resistance. If
  switch S is closed, what would happen to the
  electric potential at P and Q?
•   Potential at P Potential at Q
• A. increased increased
• B. increased decreased
• C. unchanged unchanged
• D. decreased increased
• E. decreased decreased

31

• 2000-IIA-23
• The figure shows some of the resistors in a
  network of resistors. The magnitudes and
  directions of some of the currents are marked as
  shown. Find the magnitude and direction of the
  current passing through the resistor R.
• A. 0.2 A from right to left
• B. 0.2 A from left to right
• C. 0.4 A from right to left
• D. 0.4 A from left to right
• E. It cannot be determined as the value of R is
  not given.

32

• 2001-IIA-21
• The above figure shows a network of resistors.
  If a voltage of 100 V is applied across terminals
  A and B, the potential difference between C and D
  is 80 V. If the voltage is applied across
  terminals C and D instead, what is the potential
  difference between A and
• A. 80 V B. 60 V C. 40 V D. 20 V
• E. It cannot be found as the value of R is not
  known.

33

• 2001-IIA-22
• X and Y are bulbs with ratings 6 V, 12 W and
  6 V, 3 W respectively. If they are connected to
  12 V supply of negligible internal resistance,
  which of the following connections allows the two
  bulbs to work at their respective rated values?
• A. B. C.
• D. E.

34

• 2002-IIA-24
• In the above circuit, the d.c. supply has an
  e.m.f. of 9 V and its internal resistance is
  negligible. V1, V2 and V3 are three voltmeters
  with the same finite internal resistance. If V1
  reads 4 V, find the reading of V3 and the
  potential difference across resistor R.
• Voltmeter reading of V3 Potential difference
  across R
• A. 3V 1 V
• B. 5V 1 V
• C. 1V 5 V
• D. 1V 3 V

35

• 2003-IIA-25
• In the above circuit, a battery of e.m.f. 6 V
  and negligible internal resistance is connected
  to three resistors. What are the electric
  potentials at P before and after switch S is
  closed?
• Before After
• A. 6 V 3 V
• B. 3 V 3 V
• C. 2 V 2 V
• D. 2 V 3 V

36

• 2003-IIA-27
• In the above circuit, the battery has constant
  e.m.f. and negligible internal resistance. A
  high-resistance voltmeter connected across
  terminals a and b reads 4 V. If a low-resistance
  ammeter is connected across a and b, the ammeter
  should read
• A. 0.6 A B. 1.0 A. C. 1.2
  A. D. 2.0 A.
• 2004-IIA-20
• In the above network of resistors, the
  resistance of S is infinitely large and the two
  resistors R are identical. If the equivalent
  resistance across CD is 2.5 kW, what is the
  equivalent resistance across AC?
• A. 2.5 kW B. 3.5 kW C. 5.0 kW
  D. infinitely large

37

• 2003-IIA-27
• The graph shows the I-V characteristic of two
  light bulbs X and Y, which are marked
  respectively as 200 V, 100 W and 200 V, 60 W.
  If X and Y are connected in series to 200 V mains
  supply, what is the approximate power dissipated
  in each bulb?
•   X Y
•   A. 12 W 36 W
• B. 15 W 25 W
• C. 40 W 20 W
• D. 50 W 30 W

38

• Internal resistance r
• When the p.d. across a source is measured
  by a high resistance voltmeter, the volmeter
  reading seems to with upon loading of a
  resistor.
• Unloaded source loaded source
• Reading Reading
• The in voltage is due to the cell itself
  having some resistance, known as internal
  resistance r.
• In general the terminal p.d. V when a current I
  is drawn from the source is
• Eq.(7)
• Internal resistance of a low-voltage supply such
  as the from which currents are required
  must be , while that of a high-voltage supply
  like an must be for sake.

39

• The internal resistance of a source can be
  estimated by taking several p.d. (V) and current
  (I) measurements and plotting the graph of
  against . Since
• the y-intercept and slope of the graph
  respectively corresponds to the and
  of the source.
• When a bus or car starts with the lights on, the
  lights dim a bit because an extra is
  drawn from the starter motor, which the
  potential drop across the internal resistor, and
  hence the terminal p.d., so as the
  across the lights.

40
Examination questions

• 1990-IIA-32
• The figure above shows part of a circuit which
  carries a circuit of 1 A from X to Y through a
  cell of e.m.f. 3 V and internal resistance 1 ?.
  The potential difference between X and Y is
•   A. 0V. B. 1V. C. 2V.
  D. 3V. E. 4V.
• 1993-IIA-31
• Two cells X and Y, each of internal resistance
  3?, are connected with two 6 ? as shown in the
  circuit. If cell X has e.m.f. 12 V and the
  galvanometer G shows null deflection, what is the
  e.m.f. of the cell Y?
•   A. 3.0 V B. 4.0 V C. 4.8 V D.
  6.0 V E. 7.2 V

41

• 1997-IIA-29
• A cell of e.m.f. 1.5 V is connected in series
  with a variable resistor of resistance R and an
  ammeter A of resistance 0.5 ?. By varying R, a
  series of ammeter readings, I, are taken and a
  graph of 1/I against R is plotted. The value of
  the y-intercept is found to be 0.88 A-1. What is
  the internal resistance of the cell?
•   A. 0.59 ? B. 0.82 ? C. 1.14 ? D. 1.20
  ? E. 1.32 ?

42

• Power output and efficiency
• When a load R is connected to a source of e.m.f.
  ? and internal resistance r, the power dissipated
  to R and the efficiency of power conversion
  depends on the value of .
• Current in the circuit
• Power input
• Power output
• Efficiency

43
Examination questions

• 2002-IIA-25
• A d.c. supply of constant e.m.f. and internal
  resistance is connected to a variable resistor of
  resistance R. Which of the following graphs best
  shows how the total power P delivered by the
  supply varies with R?
• A. B. C. D.
• 2004-IIA-22
• In the above circuit the battery has an
  e.m.f. of
• 6 V and internal resistance 2 W. S is a
  standard
• load resistor of 6 W and R is a rheostat. What
  will
• the setting of R be if the power delivered by
  the
• battery to the load resistor S is at a maximum?
• A. 0 W B. 2 W C. 4 W D. 6 W

44
3.3.4 Potential Divider

• Forms of a potential divider
• A potential divider provides a convenient way of
  getting a p.d. from a p.d.
• Using two fixed resistors
• The simplest form of a potential divider
  consists of two resistors in .
• Using a varaible resistors
• A or types variable resistor can be used
  to provide an output p.d. that varies more .

45

• Effect of external load resistance on the output
  voltage
• When the output terminal of a potential divider
  is loaded by an external resistor, the potential
  divider is now developed across the output
  resistor (BC) in with the load
  resistor, which the equivalent
  resistance of the resistors. The p.d. across the
  output (or load) resistor therefore .
• However, the output p.d. does not change much so
  long as the resistance of the load is very ,
  for this ensures the load does not draw much
  from the potential divider.