Chapter 8 Current Electricity Part 1 – Physics free study material by TEACHING CARE online tuition and coaching classes

 

 

Electric Current.

• Definition : The time rate of flow of charge through any cross-section is called So if through a

 

cross-section, DQ charge passes in time Dt then i

= DQ

and instantaneous current

i = Lim ΔQ = dQ . If flow is

 

av            Dt

Δt ®0 Δt          dt

 

uniform then i = Q . Current is a scalar quantity. It’s S.I. unit is ampere (A) and C.G.S. unit is emu and is called biot

t

(Bi), or ab ampere. 1A = (1/10) Bi (ab amp.)

• The direction of current : The conventional direction of current is taken to be the direction of flow of positive charge, e. field and is opposite to the direction of flow of negative charge as shown below.

 

 

i                                                                                                                                                  i

 

r                                                                                                r

E                                                                                                                                                 E

Though conventionally a direction is associated with current (Opposite to the motion of electron), it is not a vector. It is because the current can be added algebraically. Only scalar quantities can be added algebraically not the vector quantities.

• Charge on a current carrying conductor : In conductor the current is caused by electron (free electron). The no. of electron (negative charge) and proton (positive charge) in a conductor is same. Hence the net charge in a current carrying conductor is
• Current through a conductor of non-uniform cross-section : For a given conductor current does not change with change in cross-sectional In the following figure i₁ = i₂ = i₃

 

• Types of current : Electric current is of two type :

Note :@ In our houses ac is supplied at 220V, 50Hz.

(6)  Current in difference situation :

 

 

 

 

• Current carriers : The charged particles whose flow in a definite direction constitutes the electric current are called current In different situation current carriers are different.
  • Solids : In solid conductors like metals current carriers are free
  • Liquids : In liquids current carriers are positive and negative ions.
  • Gases : In gases current carriers are positive ions and free
  • Semi conductor : In semi conductors current carriers are holes and free

 

 

Current density (J).

In case of flow of charge through a cross-section, current density is defined as a vector having magnitude equal to current per unit area surrounding that point. Remember area is normal to the direction of charge flow (or

 

current passes) through that point. Current density at point P is given by

J = di n

dA

 

 

 

If the cross-sectional area is not normal to the current, the cross-sectional area normal to current in accordance with following figure will be dA cosq and so in this situation:

 

J =         di      

dA cos q

i.e. di = JdA cos q

or di = J .dA Þ i = ò J

× dA

 

 

 

 

i.e., in terms of current density, current is the flux of current density.

Note : @If current density J is uniform for a normal cross-section A then :

r          r

i = ò × ds =     × ò ds [as J = constant]

 

 

or   i = J × A

= JA cos 0 = JA   Þ

J = i

A

[as ò dA = A

J              J

and q = 0^o]

 

• Unit and dimension : Current density J is a vector quantity having I. unit Amp/m² and dimension.[L^–2A]
• Current density in terms of velocity of charge : In case of uniform flow of charge through a cross-

 

section normal to it as

i = nqvA

so,

J =  i A

n = (nqv) n

or J

= nqv

= v (r)

[With

r = charge

volume

= nq ]

 

i.e., current density at a point is equal to the product of volume charge density with velocity of charge distribution at that point.

 

• Current density in terms of electric field : Current density relates with electric field as where s = conductivity and r = resistivity or specific resistance of

 

 

• Direction of current density J is same as that of electric field E .

J = ω E = E ;

ω

 

 

• If electric field is uniform (e. E = constant ) current density will be constant [as s = constant]
• If electric field is zero (as in electrostatics inside a conductor), current density and hence current will be

 

Conduction of Current in Metals.

According to modern views, a metal consists of a ‘lattice’ of fixed positively charged ions in which billions and billions of free electrons are moving randomly at speed which at room temperature (i.e. 300 K) in accordance with

 

 

kinetic theory of gases is given by v

rms   =            =

~– 10⁵ m / s

 

 

The randomly moving free electrons inside the metal collide with the lattice and follow a zig-zag path as shown in figure (A).

 

 

 

 

 

 

However, in absence of any electric field due to this random motion, the number of electrons crossing from left to right is equal to the number of electrons crossing from right to left (otherwise metal will not remain equipotential) so the net current through a cross-section is zero.

When an electric field is applied, inside the conductor due to electric force the path of electron in general becomes curved (parabolic) instead of straight lines and electrons drift opposite to the field figure (B). Due to this drift the random motion of electrons get modified and there is a net transfer of electrons across a cross-section resulting in current.

 

 

• Drift velocity : Drift velocity is the average uniform velocity acquired by free electrons inside a metal by the application of an electric field which is responsible for current through Drift velocity is very small it is of the order of 10^–4 m/s as compared to thermal speed (~– 10⁵ m / s) of electrons at room temperature.

If suppose for a conductor

n = Number of electron per unit volume of the conductor

A = Area of cross-section

V = potential difference across the conductor

E = electric field inside the conductor

æ        1 ö

 

i = current, J = current density, r = specific resistance, s = conductivity

çs = r ÷

then current relates with

 

 

drift velocity as i = neAv

we can also write v

è          ø

=    i   = J  = ωE =   E =    V  .

 

^{d                         d}    neA    ne

ne    ωne

ω l ne

 

Note : @The direction of drift velocity for electron in a metal is opposite to that of applied electric field (i.e. current density J ).

@           vd µ E i.e., greater the electric field, larger will be the drift velocity.

@          When a steady current flows through a conductor of non-uniform cross-section drift velocity

 

ç

varies inversely with area of cross-section æv

è d

µ 1 ö

A ÷

ø

 

@           If diameter of a conductor is doubled, then drift velocity of electrons inside it will not change.

• Relaxation time (t) : The time interval between two successive collisions of electrons with the positive

 

ions in the metallic lattice is defined as relaxation time

t =        mean free path         =   l  

with rise in

 

 

temperature v_rms increases consequently t decreases.

r.m.s. velocity of electrons

vrms

 

• Mobility : Drift velocity per unit electric field is called mobility of electron e. m = vd

E

. It’s unit is

m2         .

volt – sec

 

 

 

 

 

Example: 1         The potential difference applied to an X-ray tube is 5 KV and the current through it is 3.2 mA. Then the number of electrons striking the target per second is                                                        [IIT-JEE (Screening) 2002]

(a) 2 ´ 10¹⁶                       (b) 5 ´ 10⁶                           (c) 1 ´ 10¹⁷                    (d) 4 ´ 10¹⁵

 

 

Solution : (a)

i = q

t

= ne t

Þ n = it =

e

3.2 ´ 10 ^–3 ´ 1

 

1.6 ´ 10 ^–19

= 2 ´ 10¹⁶

 

Example: 2         A beam of electrons moving at a speed of 10⁶ m/s along a line produces a current of 1.6 ´ 10^–6 A. The number of electrons in the 1 metre of the beam is                                                                                                          [CPMT 2000]

(a) 10⁶                             (b) 10⁷                                (c) 10¹³                         (d) 10¹⁹

 

q          q          qv

 

 

nev

 

ix          1.6 ´ 10^–6 ´ 1           ₇

 

Solution : (b)

i = t

=              =        =

(x / v)       x          x

Þ   n = ev = 1.6 ´ 10^–19 ´ 10⁶ = 10

 

Example: 3         In the Bohr’s model of hydrogen atom, the electrons moves around the nucleus in a circular orbit of a

radius 5 ´ 10^–11 metre. It’s time period is 1.5 ´ 10^–16 sec. The current associated is                           [MNR 1992]

(a) Zero                           (b) 1.6 ´ 10^–19 A                         (c) 0.17 A                            (d) 1.07 ´ 10^–3 A

 

 

 

q      1.6 ´ 10 ^–19                   _–3

 

Solution : (d)

i = T = 1.5 ´ 10 ^–16

= 1.07 ´ 10      A

 

Example: 4         An electron is moving in a circular path of radius 5.1 ´ 10^–11 m at a frequency of 6.8 ´ 10¹⁵ revolution/sec. The

equivalent current is approximately                                                 [MP PET 2000 Similar to EAMCET (Med.) 2000]

(a) 5.1 ´ 10^–3 A                         (b) 6.8 ´ 10^–3 A                             (c) 1.1 ´ 10^–3 A                     (d) 2.2 ´ 10^–3 A

 

 

Solution : (c)

n = 6.8 ´ 10¹⁵

Þ T =

1

6.8 ´ 10¹⁵

sec

Þ i = Q = 1.6 ´ 10 ^–19 ´ 6.8 ´ 10¹⁵ = 1.1 ´ 10^–3 A

T

 

Example: 5         A copper wire of length 1m and radius 1mm is joined in series with an iron wire of length 2m and radius 3mm and a current is passed through the wire. The ratio of current densities in the copper and iron wire is

[MP PMT 1994]

(a) 18 : 1                         (b) 9 : 1                              (c) 6 : 1                        (d) 2 : 3

 

i                                                                    1             J

A        æ r ö ²

æ 3 ö ²     9

 

Solution : (b)       We know J =                           when i = constant

J µ          Þ

    c   =     i    = ç i ÷

= ç   ÷   =

 

A                                                                   A                 J_i

Ac           è rc ø

è 1 ø         1

 

Example: 6         A conducting wire of cross-sectional area 1 cm² has 3 ´ 10²³ m^–3 charge carriers. If wire carries a current of 24 mA, the drift speed of the carrier is                                                                                       [UPSEAT 2001]

(a) 5 ´ 10^–6 m/s                             (b) 5 ´ 10^–3 m/s                         (c) 0.5 m/s                             (d) 5 ´ 10^–2 m/s

i                           24 ´ 10 ^–3                            _–3

 

Solution : (b)

vd   = neA = 3 ´ 10²³ ´ 1.6 ´ 10 ^–19 ´ 10 ^–4

= 5 ´ 10

m / s

 

Example: 7         A wire has a non-uniform cross-sectional area as shown in figure. A steady current i flows through it. Which

one of the following statement is correct

• The drift speed of electron is constant (b) The drift speed increases on moving from A to B

(c) The drift speed decreases on moving from A to B                 (d) The drift speed varies randomly

Solution : (c)        For a conductor of non-uniform cross-section vd µ                      1              

Area of cross – section

Example: 8 In a wire of circular cross-section with radius r, free electrons travel with a drift velocity v, when a current i flows through the wire. What is the current in another wire of half the radius and of the some material when the drift velocity is 2v                                                                                                                                                                [MP PET 1997]

(a) 2i                                        (b) i                                               (c) i/2                          (d) i/4

æ r ö ²          nepr ²v       i

 

Solution : (c)

i = neAv_d = nepr²v and i‘ = nep ç   ÷ .2v =                   =

 

è 2 ø                    2         2

Example: 9         A potential difference of V is applied at the ends of a copper wire of length l and diameter d. On doubling only

d, drift velocity                                                                                                                     [MP PET 1995]

• Becomes two times (b) Becomes half                   (c) Does not change         (d) Becomes one fourth

 

 

 

Solution : (c)        Drift velocity doesn’t depends upon diameter.

Example: 10       A current flows in a wire of circular cross-section with the free electrons travelling with a mean drift velocity v.

If an equal current flows in a wire of twice the radius new mean drift velocity is

 

 

(a) v                                          (b)

v                                             (c)

2

v                                    (d) None of these

4

 

Solution : (c)        By using vd   =   i

neA

Þ vd  µ 1

A

Þ v‘ = v

4

 

Example: 11       Two wires A and B of the same material, having radii in the ratio 1 : 2 and carry currents in the ratio 4 : 1. The ratio of drift speeds of electrons in A and B is

(a) 16 : 1                         (b) 1 : 16                            (c) 1 : 4                        (d) 4 : 1

 

i₁      A₁     vd

 

 

r ² vd

vd             16

 

Solution : (a)       As i = neA vd   Þ       =

´    1    = 1 .   1    Þ    1    =

 

r

d

i2       A2      v 2

2   v            v            1

2

d

d

2                   2

 

Tricky example: 1

In a neon discharge tube 2.9 ´ 1018 Ne+ ions move to the right each second while 1.2 ´ 1018 electrons move to the left per second. Electron charge is 1.6 ´ 10–19 C. The current in the discharge tube [MP PET 1999] (a) 1 A towards right               (b) 0.66 A towards right        (c) 0.66 A towards left (d) Zero Solution: (b)      Use following trick to solve such type of problem. Trick : In a discharge tube positive ions carry q units of charge in t seconds from anode to cathode and negative carriers (electrons) carry the same amount of charge from cathode to anode in t¢ second. The current in the tube is i = q + q’ . t     t’ Hence in this question current i = 2.9 ´ 1018 ´ e + 1.2 ´ 1018 ´ e = 0.66 A towards right. 1                      1

Tricky example: 2

If the current flowing through copper wire of 1 mm diameter is 1.1 amp. The drift velocity of electron is (Given density of Cu is 9 gm/cm3, atomic weight of Cu is 63 grams and one free electron is contributed by each atom)                                                                                                                          [J&K CEET 2000] (a) 0.1 mm/sec                               (b) 0.2 mm/sec                          (c) 0.3 mm/sec                      (d) 0.5 mm/sec Solution: (a) 6.023 ´ 1023 atoms has mass = 63 ´ 10–3 kg So no. of atoms per m3 = n = 6.023 ´ 1023 ´ 9 ´ 103 = 8.5 ´ 1028 63 ´ 10 –3 vd   =    i     =                              1.1                            = 0.1 ´ 10 –3 m / sec = 0.1mm / sec neA       8.5 ´ 1028 ´ 1.6 ´ 10 –19 ´ p ´ (0.5 ´ 10 –3 )2

Ohm’s Law.

 

 

 

If the physical circumstances of the conductor (length, temperature, mechanical strain etc.) remains constant, then the current flowing through the conductor is directly proportional to the potential difference across it’s two ends

i.e. i µ V

 

 

Þ V = iR

or V = R ; where R is a proportionality constant, known as electric resistance.

i

 

• Ohm’s law is not a universal law, the substance which obeys ohm’s law are known as ohmic substance for such ohmic substances graph between V and i is a straight line as At different temperatures V-i curves are different.

 

 

 

 

 

 

 

• The device or substances which doesn’t obey ohm’s law g. gases, crystal rectifiers, thermoionic valve, transistors etc. are known as non-ohmic or non-linear conductors. For these V-i curve is not linear. In these situation the ratio between voltage and current at a particular voltage is known as static resistance. While the rate of change of voltage to change in current is known as dynamic resistance.

 

V

Rst = i

=   1 tan q

 

 

while

Rdyn

= DV

DI

=   1 tan f

 

• Some other non-ohmic graphs are as follows :

 

Resistance.

• Definition : The property of substance by virtue of which it opposes the flow of current through it, is known as the
• Cause of resistance of a conductor : It is due to the collisions of free electrons with the ions or atoms of the conductor while drifting towards the positive end of the

 

 

• Formula of resistance : For a conductor if l = length of a conductor A = Area of cross-section of conductor, n = of free electrons per unit volume in conductor, t = relaxation time then resistance of

conductor R = ω l = m . l ; where r = resistivity of the material of conductor

A    ne ²t    A

• Unit and dimension : It’s I. unit is Volt/Amp. or Ohm (W). Also 1 ohm = 1volt= 108 emu of potential =

 

 

10⁹ emu of resistance. It’s dimension is [ML²T ^–3 A^–2 ] .

1Amp

10 ^-1 emu of current

 

 

• Conductance (C) : Reciprocal of resistance is known as

C = 1

R

It’s unit is

1 or W^–1 or

W

 

“Siemen”.

i

 

Slope = tanq = i

V

 

= 1 = C R

 

 

 

• Dependence of resistance : Resistance of a conductor depends on the following
  • Length of the conductor : Resistance of a conductor is directly proportional to it’s length e. R µ l e.g. a

conducting wire having resistance R is cut in n equal parts. So resistance of each part will be R .

n

• Area of cross-section of the conductor : Resistance of a conductor is inversely proportional to it’s area of

 

cross-section i.e.

R µ 1

A

 

 

 

• Material of the conductor : Resistance of conductor also depends upon the nature of material e.

for different conductors n is different. Hence R is also different.

R µ 1 ,

n

 

• Temperature : We know that

R =    m   . l

ne²t    A

Þ R µ l

t

when a metallic conductor is heated, the atom in

 

the metal vibrate with greater amplitude and frequency about their mean positions. Consequently the number of collisions between free electrons and atoms increases. This reduces the

relaxation time t and increases the value of resistance R i.e. for a conductor

Resistance µ temperature .

If     R₀ = resistance of conductor at 0^oC R_t = resistance of conductor at t^oC

 

and a, b = temperature co-efficient of resistance (unit ® per^oC)

O                               t^oC

 

 

then R

= R (1 + at + bt ² ) for t > 300^oC and R = R (1 + αt) for t £ 300^oC     or

a = Rt – R0

 

 

t              0                                                                      t          0

R0 ´ t

 

 

 

Note : @               If R and R are the resistances at t ^oC and t ^oC respectively then

R1 = 1 + a t1 .

 

 

 

1              2                                               1                 2

R2       1 + a t2

 

@          The value of a is different at different temperature. Temperature coefficient of resistance

 

1             2

averaged over the temperature range t ^oC to t ^oC is given by a =

R2 – R1

which gives R

= R  [1 +

 

R1(t2 – t1)

2          1

a (t₂ – t₁)]. This formula gives an approximate value.

• Resistance according to potential difference : Resistance of a conducting body is not unique but depends on it’s length and area of cross-section e. how the potential difference is applied. See the following figures

 

 

Length = b

Area of cross-section = a ´ c

Length = a

Area of cross-section = b ´ c

Length = c

Area of cross-section = a ´ b

 

æ   b    ö

Resistance R = r ç     ÷

æ   a    ö

Resistance R = r ç     ÷

æ   c     ö

Resistance R = r ç     ÷

 

è a ´ c ø

è b ´ c ø

è a ´ b ø

 

(7)  Variation of resistance of some electrical material with temperature :

• Metals : For metals their temperature coefficient of resistance a > So resistance increases with temperature.

Physical explanation : Collision frequency of free electrons with the immobile positive ions increases

• Solid non-metals : For these a = 0. So resistance is independence of

Physical explanation : Complete absence of free electron.

• Semi-conductors : For semi-conductor a < 0 e. resistance decreases with temperature rise.

Physical explanation : Covalent bonds breaks, liberating more free electron and conduction increases.

• Electrolyte : For electrolyte a < 0 e. resistance decreases with temperature rise.

Physical explanation : The degree of ionisation increases and solution becomes less viscous.

• Ionised gases : For ionised gases a < 0 e. resistance decreases with temperature rise.

Physical explanation : Degree of ionisation increases.

• Alloys : For alloys a has a small positive So with rise in temperature resistance of alloys is almost constant. Further alloy resistances are slightly higher than the pure metals resistance.

Alloys are used to made standard resistances, wires of resistance box, potentiometer wire, meter bridge wire etc. Commonly used alloys are : Constantan, mangnin, Nichrome etc.

• Super conductors : At low temperature, the resistance of certain substances becomes exactly (e.g. Hg

below 4.2 K or Pb below 7.2 K).

 

 

These substances are called super conductors and phenomenon super conductivity. The temperature at which resistance becomes zero is called critical temperature and depends upon the nature of substance.

 

 

Resistivity or Specific Resistance (r).

 

• Definition : From

R = r l ;

A

If l = 1m, A = 1 m² then

R = q

i.e. resistivity is numerically equal to the

 

resistance of a substance having unit area of cross-section and unit length.

• Unit and dimension : It’s I. unit is ohm ´ m and dimension is [ML³T ^–3 A^–2]

 

 

(3)   It’s formula :

r =    m ne ²t

 

• It’s dependence : Resistivity is the intrinsic property of the It is independent of shape and size of the body (i.e. l and A). It depends on the followings :
  • Nature of the body : For different substances their resistivity also different g. r_silver = minimum =

1.6 ´ 10^–8 W-m and r_{fused quartz} = maximum » 10¹⁶ W-m

• Temperature : Resistivity depends on the For metals r_t = r₀ (1 + aDt) i.e. resitivity increases with temperature.

r increases with temperature                                     r decreases with temperature                               r decreases with temperature and

becomes zero at a certain temperature

 

• Impurity and mechanical stress : Resistivity increases with impurity and mechanical
• Effect of magnetic field : Magnetic field increases the resistivity of all metals except iron, cobalt and
• Effect of light : Resistivity of certain substances like selenium, cadmium, sulphides is inversely proportional to intensity of light falling upon

 

(5)  Resistivity of some electrical material :

ωinsulator

(Maximum for fused quartz)

• ωalloy >ωsemi-conductor

• ωconductor

(Minimum for silver)

 

Note : @Reciprocal of resistivity is called conductivity (s) i.e.

[M -1 L-3 T 3 A 2 ] .

= 1

s

r

with unit mho/m and dimensions

 

 

Stretching of Wire.

 

 

 

If a conducting wire stretches, it’s length increases, area of cross-section decreases so resistance increases but volume remain constant.

Suppose for a conducting wire before stretching it’s length = l₁, area of cross-section = A₁, radius = r₁,

A

diameter = d , and resistance R = r l1

1                                     1

1

 

Before stretching                             After stretching

 

After stretching length = l , area of cross-section = A , radius = r , diameter = d

and resistance = R

= r l2

 

 

2

 

 

R     l     A     æ l ö2

2

 

 

æ A ö2

2

 

 

A

æ r ö4

2                                        2

2

 

æ d ö4

 

Ratio of resistances

    1  =  1  ´    2  = ç 1 ÷

= ç   ² ÷

= ç ² ÷

= ç  ² ÷

 

R2       l2

A1      ç l2 ÷     ç A1 ÷

ç r1 ÷      ç d1 ÷

 

è    ø      è

R        æ l ö2

ø      è    ø      è     ø

 

• If length is given then

R µ l ² Þ ¹

R2

= ç ¹ ÷

è l2 ø

 

1     R        æ r ö4

 

• If radius is given then R µ

r

Þ   1

4

R2

= ç ² ÷

è r1 ø

 

Note : @               After stretching if length increases by n times then resistance will increase by n² times i.e.

 

R = n²R . Similarly if radius be reduced to 1 times then area of cross-section decreases 1

times

 

2             1                                                                         n                                                                                            n 2

 

so the resistance becomes n⁴ times i.e.

R2 = n⁴ R1 .

 

@          After stretching if length of a conductor increases by x% then resistance will increases by 2x % (valid only if x < 10%)

 

 

Various Electrical Conducting Material For Specific Use.

• Filament of electric bulb : Is made up of tungsten which has high resistivity, high melting
• Element of heating devices (such as heater, geyser or press) : Is made up of nichrome which has high resistivity and high melting
• Resistances of resistance boxes (standard resistances) : Are made up of manganin, or constantan as these materials have moderate resistivity which is practically independent of temperature so that the specified value of resistance does not alter with minor changes in
• Fuse-wire : Is made up of tin-lead alloy (63% tin + 37% lead). It should have low melting point and high It is used in series as a safety device in an electric circuit and is designed so as to melt and thereby open

 

 

the circuit if the current exceeds a predetermined value due to some fault. The function of a fuse is independent of its length.

Safe current of fuse wire relates with it’s radius as i µ r ^3/2 .

• Thermistors : A thermistor is a heat sensitive resistor usually prepared from oxides of various metals such as nickel, copper, cobalt, iron etc. These compounds are also semi-conductor. For thermistors a is very high which may be positive or The resistance of thermistors changes very rapidly with change of temperature.

i

 

 

 

 

 

Thermistors are used to detect small temperature change and to measure very low temperature.

 

Example: 12       Two wires of resistance R₁ and R₂ have temperature co-efficient of resistance a₁ and a₂ respectively. These are joined in series. The effective temperature co-efficient of resistance is                                                            [MP PET 2003]

 

 

(a)

a₁ + a ₂

2

(b)

(c)

a₁ R1 + a ₂ R2

R1 + R2

(d)

 

 

 

Solution : (c)        Suppose at t^oC resistances of the two wires becomes R1t

and R2t

respectively and equivalent resistance

 

becomes R_t. In series grouping R_t = R_1t + R_2t, also R_1t = R₁(1 + a₁t) and R_2t = R₂(1 + a₂t)

 

 

R = R (1 + a t) + R (1 + a t) = (R

+ R ) + (R a

+ R a )t = (R   +

é + R1a₁ + R2a ₂ tù .

 

 

t             1             1            2             2

1         2            1   1         2 2

1     R2 )ê1

ë

R1 + R2         ú

 

û

Hence effective temperature co-efficient is

R1a1 + R2a 2 .

R1 + R2

 

Example: 13       From the graph between current i & voltage V shown, identity the portion corresponding to negative resistance

[CBSE PMT 1997]

• DE
• CD
• BC
• AB

 

 

 

 

Solution : (b)

R = DV , in the graph CD has only negative slope. So in this portion R is negative.

DI

 

Example: 14       A wire of length L and resistance R is streched to get the radius of cross-section halfed. What is new resistance

[NCERT 1974; CPMT 1994; AIIMS 1997; KCET 1999; Haryana PMT 2000; UPSEAT 2001]

(a) 5 R                                     (b) 8 R                                          (c) 4 R                                 (d) 16 R

 

R        æ r ö ⁴

R       æ r / 2 ö ⁴

 

Solution : (d)       By using

    1 = ç 2 ÷

Þ           = ç         ÷      Þ   R‘ = 16R

 

 

R2       è r1 ø

R ‘    è   r   ø

 

Example: 15       The V–i graph for a conductor at temperature T₁ and T₂ are as shown in the figure. (T₂ – T₁) is proportional to

 

• cos 2q
• sinq
• cot 2q
• tanq

Solution : (c)        As we know, for conductors resistance µ Temperature.

From figure R₁ µ T₁ Þ tanq µ T₁ Þ tanq = kT₁             ……. (i) (k = constant) and R₂ µ T₂ Þ tan (90^o – q) µ T₂ Þ cotq = kT₂……………………………….. (ii)

From equation (i) and (ii) k(T₂ – T₁) = (cot q – tanq )

 

(T – T  ) = æ cosq – sinq ö = (cos ² q – sin ² q ) =  cos 2q  = 2 cot 2q

 

Þ (T

– T ) µ cot 2q

 

è

ø

2        1       ç sinq

cosq ÷

sinq cosq

sinq cosq                          ^{2     1}

 

Example: 16       The resistance of a wire at 20^oC is 20 W and at 500^oC is 60W. At which temperature resistance will be 25W

[UPSEAT 1999]

(a) 50^oC                                   (b) 60^oC                                       (c) 70^oC                               (d) 80^oC

 

Solution : (d)       By using

R1 = (1 + a t1 ) Þ

20 = 1 + 20a

Þ a =   1 

 

R2       (1 + a t 2 )

60     1 + 500a

220

æ1 + 1 ´ 20ö

 

20     ç        220        ÷

 

Again by using the same formula for 20W and 25W Þ

= è                         ø

25      æ1 +  1 ´ t ö

Þ t = 80^oC

 

è

ø

ç       220     ÷

Example: 17       The specific resistance of manganin is 50 ´ 10^–8 Wm. The resistance of a manganin cube having length 50 cm

is                                                                                                                                      [MP PMT 1978]

(a) 10^–6 W                                  (b) 2.5 ´ 10^–5 W                            (c) 10^–8 W                              (d) 5 ´ 10^–4 W

 

 

Solution : (a)

R = r l   =

A

50 ´ 10 ^–8 ´ 50 ´ 10 ^–2

(50 ´ 10 ^–2 )²

= 10 ^–6 W

 

Example: 18       A rod of certain metal is 1 m long and 0.6 cm in diameter. It’s resistance is 3 ´ 10^–3W. A disc of the same metal is 1 mm thick and 2 cm in diameter, what is the resistance between it’s circular faces.

(a) 1.35 ´ 10^–6W                     (b) 2.7 ´ 10^–7 W                            (c) 4.05 ´ 10^–6W                 (d) 8.1 ´ 10^–6 W

 

l    Rdisc

 

ldisc

Arod

Rdisc

10 -3

p (0.3 ´ 10 ^–2 )²                                _–7

 

Solution : (b)       By using

R = r.    ;

A

Rrod

=

l rod

´

Adisc

Þ                    =            ´

3 ´ 10 ^–3         1

p (10 ^–2 )²

Þ R_disc = 2.7 ´ 10 W.

 

 

 

Example: 19       An aluminium rod of length 3.14 m is of square cross-section 3.14 ´ 3.14 mm². What should be the radius of 1 m long another rod of same material to have equal resistance

• 2 mm (b) 4 mm                                      (c) 1 mm                             (d) 6 mm

 

 

Solution : (c)        By using

R = r. l

A

Þ l µ A Þ

3.14

 

1

= 3.14 ´ 3.14 ´ 10 ^–6

p ´ r ²

Þ r = 10^–3

m = 1 mm

 

Example: 20       Length of a hollow tube is 5m, it’s outer diameter is 10 cm and thickness of it’s wall is 5 mm. If resistivity of the material of the tube is 1.7 ´ 10^–8 W´m then resistance of tube will be

(a) 5.6 ´ 10^–5 W                        (b) 2 ´ 10^–5 W                               (c) 4 ´ 10^–5 W                        (d) None of these

 

Solution : (a)       By using R = r l

 

here A = p (r ² – r ² )

 

 

Outer radius r₂

. A ;                    2       1

= 5cm                                                                                                 r2

 

Inner radius r₁ = 5 – 0.5 = 4.5 cm

 

So R = 1.7 ´ 10 ^–8 ´

5

p {(5 ´ 10 ^–2 )² – (4.5 ´ 10 ^–2 )² }

= 5.6 ´ 10 ^–5 W

 

Example: 21       If a copper wire is stretched to make it 0.1% longer, the percentage increase in resistance will be

[MP PMT 1996, 2000; UPSEAT 1998; MNR 1990]

(a) 0.2                             (b) 2                                  (c)  1                            (d) 0.1

 

 

Solution : (a)       In case of streching R µ l²             So

DR = 2 Dl = 2 ´ 0.1 = 0.2

 

 

R            l

Example: 22           The  temperature  co-efficient  of  resistance  of  a  wire  is  0.00125/^oC.  At  300  K.  It’s  resistance  is  1W.  The resistance of the wire will be 2W at                                                                               [MP PMT  2001; IIT 1980]

(a) 1154 K                               (b) 1127 K                                   (c) 600 K                             (d) 1400 K

 

 

Solution: (b)        By using R = R

(1 + aDt) Þ

R1 = 1 + a t1

 

So 1 = 1 + (300 – 273)a

2

Þ t = 854^oC = 1127 K

 

t              o

2

1 + a t 2               2

1 + a t 2

 

R

Example: 23       Equal potentials are applied on an iron and copper wire of same length. In order to have same current flow in

 

æ

the wire, the ratio ç

riron

ö

÷ of their radii must be [Given that specific resistance of iron = 1.0 ´ 10^–7

Wm and

 

ç rcopper ÷

that of copper = 1.7 ´ 10–8 Wm]     [MP PMT 2000]   Solution: (b) (a) About 1.2 V = constant., i = constant. (b) About 2.4 So R = constant (c) About 3.6 (d) About 4.8

è             ø

 

 

 

Þ     Pili    = r CulCu

 

 

ri

Þ r _ili

 

 

=  r CulCu

 

 

Þ   ri      =                =

 

 

 

 

=              » 2.4

 

Ai                 ACu

2              2

Cu

rCu

 

r

Example: 24       Masses of three wires are in the ratio 1 : 3 : 5 and their lengths are in the ratio 5 : 3 : 1. The ratio of their electrical resistance is                                                                                                               [AFMC 2000]

(a) 1 : 3 : 5                      (b) 5 : 3 : 1                         (c) 1 : 15 : 125              (d) 125 : 15 : 1

l           l ²        l ²             æ             m ö

 

Solution: (d)

R = r     = r

A           V

= r     s

m

çQs = V ÷

 

è                 ø

 

 

 

l ²     l ²      l ²             9 1

 

R1 : R2 : R3 =   ¹ : ² : ³   = 25 :      :

= 125 : 15 : 1

 

m1     m2      m3                3 5

Example: 25  Following figure shows cross-sections through three long conductors of the same length and material, with square cross-section of edge lengths as shown. Conductor B will fit snugly within conductor A, and conductor C will fit snugly within conductor B. Relationship between their end to end resistance is

 

 

 

• R_A = R_B = R_C
• R_A > R_B > R_C
• R_A < R_B < R
• Information is not sufficient

Solution : (a)       All the conductors have equal lengths. Area of cross-section of A is {( Similarly area of cross-section of B = Area of cross-section of C = a²

 

 

 

 

3 a)² – (

 

 

2 a)² }= a ²

 

 

Hence according to formula

R = r

l ; resistances of all the conductors are equal i.e. R

A                                                                                               A

= R_B

= R_C

 

 

Example: 26            Dimensions of a block are 1 cm ´ 1 cm ´ 100 cm. If specific resistance of its material is 3 ´ 10^–7 ohm–m, then the resistance between it’s opposite rectangular faces is                                                                                   [MP PET 1993]

(a) 3 ´ 10^–9 ohm                     (b) 3 ´ 10^–7 ohm                        (c) 3 ´ 10^–5 ohm                 (d) 3 ´ 10^–3 ohm

Solution: (b)        Length l = 1 cm = 10 ^–2 m

Area of cross-section A = 1 cm ´ 100 cm

= 100 cm² = 10^–2 m²

10 -2

 

Resistance R = 3 ´ 10^–7 ´              = 3 ´ 10^–7 W

10 – 2

1 cm

 

 

Note :@ In the above question for calculating equivalent resistance between two opposite square faces.

 

 

l = 100 cm = 1 m, A = 1 cm² = 10^–4 m², so resistance R = 3 ´ 10^–7 ´

1

10 -4

= 3 ´ 10^–3 W

 

 

 

 

 

 

Colour Coding of Resistance.

The resistance, having high values are used in different electrical and electronic circuits. They are generally made up of carbon, like 1 kW, 2 kW, 5 kW etc. To know the value of resistance colour code is used. These code are printed in form of set of rings or strips. By reading the values of colour bands, we can estimate the value of resistance.

The carbon resistance has normally four coloured rings or strips say A, B, C and D as shown in following figure.

 

 

 

 

Colour band A and B indicate the first two significant figures of resistance in ohm, while the C band gives the decimal multiplier i.e. the number of zeros that follows the two significant figures A and B.

Last band (D band) indicates the tolerance in percent about the indicated value or in other ward it represents the percentage accuracy of the indicated value.

The tolerance in the case of gold is ± 5% and in silver is ± 10%. If only three bands are marked on carbon resistance, then it indicate a tolerance of 20%.

The following table gives the colour code for carbon resistance.

 

Letters as an aid to memory	Colour	Figure (A, B)	Multiplier (C)	Colour	Tolerance (D)
B	Black	0	10o	Gold	5%
B	Brown	1	101	Silver	10%
R	Red	2	102	No-colour	20%
O	Orange	3	103
Y	Yellow	4	104
G	Green	5	105
B	Blue	6	106

 

 

V	Violet	7	107
G	Grey	8	108
W	White	9	109

 

Note : @           To remember the sequence of colour code following sentence should kept in memory.

B B R O Y Great Britain Very Good Wife.

 

Grouping of Resistance.

 

Series	Parallel
(1)                         R1                 R2              R3 V1                 V2               V3 i     + V – (2) Same current flows through each resistance but potential difference distributes in the ratio of resistance i.e. V µ R Power consumed are in the ratio of their resistance i.e.  P µ R Þ P1 : P2 : P3 = R1 : R2 : R3     (3) Req = R1 + R2 + R3 equivalent resistance is greater than the maximum value of resistance in the	(1)                                                                        i1             R   i2            1 i3      R2 i R3   V (2)     Same potential difference appeared across each resistance but current distributes in the reverse ratio of their resistance i.e. i µ 1 R Power consumed are in the reverse ratio of resistance i.e. P µ 1 Þ P1 : P2 : P3 = 1 : 1 : 1 R                                   R1     R2      R3 (3)   1   =   1 + 1 + 1   or Req   = (R –1 + R –1 + R –1 )–1 Req            R1       R2       R3                            1           2           3
combination.       (4)     For two resistance in series Req = R1 + R2     æ R‘ ö (5)     Potential difference across any resistance V ‘ = ç           ÷ × V ç Req  ÷ è         ø Where R¢ = Resistance across which potential difference is to be calculated, Req = equivalent resistance of that line in which R¢ is connected, V = p.d. across that line in which R¢ is connected	or R     =               R1 R2 R3                 equivalent resistance eq           R1 R2 + R2 R3 + R2 R1 is smaller than the minimum value of resistance in the combination. (4)   For two resistance in parallel R      =     R1 R2     = Multiplication eq       R1 + R2            Addition (5)   Current through any resistance i ‘ = i ´ éê Resistance of opposite branch ùú ë             Total resistance             û Where i¢ = required current (branch current) i = main current

 

 

 

 

Note : @In case of resistances in series, if one resistance gets open, the current in the whole circuit become zero and the circuit stops working. Which don’t happen in case of parallel gouging.

@          Decoration of lightning in festivals is an example of series grouping whereas all household appliances connected in parallel grouping.

@          Using n conductors of equal resistance, the number of possible combinations is 2^{n – 1}.

@          If the resistance of n conductors are totally different, then the number of possible combinations will be 2ⁿ.

Methods of Determining Equivalent Resistance For Some Difficult Networks.

• Method of successive reduction : It is the most common technique to determine the equivalent resistance. So far, we have been using this method to find out the equivalent This method is applicable only when we are able to identify resistances in series or in parallel. The method is based on the simplification of the circuit by successive reduction of the series and parallel combinations. For example to calculate the equivalent resistance between the point A and B, the network shown below successively reduced.

 

 

 

 

 

 

 

 

 

 

 

• Method of equipotential points : This method is based on identifying the points of same potential and joining The basic rule to identify the points of same potential is the symmetry of the network.
  • In a given network there may be two axes of

 

 

• Parallel axis of symmetry, that is, along the direction of current
• Perpendicular axis of symmetry, that is perpendicular to the direction of flow of

For example in the network shown below the axis AA¢ is the parallel axis of symmetry, and the axis BB¢ is the perpendicular axis of symmetry.

 

 

 

• Points lying on the perpendicular axis of symmetry may have same potential. In the given network, point 2, 0 and 4 are at the same