1.ELECTRICAL CURRENT AND OHM'S LAW
1.1 Electron Drift Velocity
Suppose that in a conductor, the number of free electrons available per m3 of the conductor material is n and let their axial drift velocity be v metres/second. In time dt, distance travelled would be v x dt. If A.is area of cross-section of the conductor, then the volume is and the number of electrons contained in this volume is vA dt. Obviously, all these electrons will cross the conductor cross-section in time dt. If e is the charge of each electron, then total charge which crosses the section in time dt is d q = dt. Since current is the rate of flow of charge, it is given as = - = :. dt . I = dt dt Current density J = ilA =ne v ampere l metre 2 Assuming a normal current density J =1.55 X 106 Alm 2 , n = 1029for a copper conductor 6 -19 and e =1. x 10 coulomb, we get 1.55 x 106 = 1029x 1.6 X 10-19x v :. v = 9.7 X 10-5m/s = 0.58 cm/min. It is seen that contrary to the common but mistaken view, the electron drift velocity is rather very slow and is independent of the current flowing and the area of the conductor. N.H.Current density i.e., the current per unit area. is a vector quantity. It is denoted by the symbol J . --> Therefore. in vector notation, the relationship between current I and J is: -->--> --> I = J. a [where a is the vector notation for area 'a'] For extending the scope of the above relationship. so that it becomes applicable for area of any shape, wewrite: f --> --> J = J .d a The magnitude of the current density can. therefore.be written as J.a. Example 1.1. A conductor material has a free- electron density of J{j 4 electrons per metel. When a voltage is applied, a constant drift velocity of 1.5 X ]0-2 metre/second is attained by the electrons. If the cross- sectional area of the material is 1 cm^2,calculate the magnitude of the current.
Electronic charge is 1.6 x ]0-19coulomb. (Electrical Engineering. Aligarh Muslim University 1981) Solution. The magnitude of the current is = amperes n = 1024;A = 1cm2 = 10-4m2 e = 1.6 x 10-19C ; v = 1.5 X 10-2m/s
= 1024X 10-4x 1.6 X 10-19x 1.5 X 10-2=0.24 A Here,..
1.2. Charge Velocity and Velocity of Field Propagation.
The speed with which charge drifts in a conductor is called the velocity of charge. As seen from above. its value is quite low, typically fraction of a metre per second.
However. the speed with which the effect of e.m.f. is experienced at all parts of the conductor resulting in the flow of current is called the velocity of propagation of electrical field. It is indepen- . 8
dent of current and voltage and has high but constant value of nearly 3 x 10.
Example 1.2. Find the velocity of charge leading to 1 A current which flows in a copper conductor of cross-section 1 em 2 and length 10 km. Free electron denSity of copper =8.5.x 1028per m3. How long will it take the electric charge to travel from one end of the conductor to the other. Solution. i =ne Av or v =ih,(' A .. v = 1/(85 y 10 X) x 1.6 x 10 1'1X (I X 10-4) = 7.35 X 10-7 mils =0.735 m ls Time taken by the charge 10lravd conductor length of 10 kIn is t = distance _ 10x 103 =1.36 x 1010S velocity 7.35 x 10-7
Now. I year::: 365 x 24 x 3600 = 31,536.000 s 10 t = 1.36 x 10 /31.536.000 = 431 years
1.3. The Idea of Electric Potential. In Fig. 1.1is shown a simple voltaic cell. It consists of copper plate (known as anode) and a zinc rod (I.e. cathode) immersed in dilute sulphuric acid (H2 So 4) contained in a suitable vessel. The chemical action taking place within the cell causes the electrons to be removed from Cu plate and to be deposited on the zinc rod at the same time. This transfer of electrons is accomplished through the agency of the diluted (H2 So 4)which is known as the electrolyte. The result is that zinc rod becomes negative due to the deposition of electrons on it and the Cu plate becomes positive due to the removal
of electrons from it. The large number of electrons collected on the zinc rod is being attracted by
anode but is prevented from returning to it by the force set up by the chemical action within the cell.
But if the two electrodes are joined by a wire externally. then electrons rush to the anode thereby equalising the charges of the two electrodes. However. due to the continuity of chemical action. a continuous difference in the number of electrons on the two electrodes is maintained which keeps up a continuous flow of current through the external circuit. The action of an electric cell is similar to that of a water pump which. while working. maintains a continuous flow of water i.e. water current through the pipe (Fig. 1.2).
It should be particularly noted that the direction of electronic current is from zinc to copper in the external circuit. However, the direction of conventional current (which is given by the direction of flow of positive charge) is from Cu to zinc. In the present case, there is no flow of positive charge as such from one electrode to another. But we can look upon the arrival of electrons on copper plate (with subsequent decrease in its positive charge) as equivalent to an actual departure of positive charge from it. When zinc is negatively charged, it is said to be at negative potential with respect to the electrolyte, whereas anode is said to be at positive potential relative to the electrolyte. Between themselves, Cu plates assumed to be at a higher potential than the zinc rod. The difference in potential is continuously maintained by the chemical action going on in the cell which supplies energy to establish this potential difference.
1.4. Resistance.
It may be defined as the property of a substance due to which it opposes (or restricts) the flow of electricity (i.e., electrons) through it. Metals (as a class), acids and salts solutions are good conductors of electricity. Amongst pure metals,silver, copper and aluminium are very good conductors in the given order.* This, as discussed earlier, is due to the presence of a large number of free or loosely - attached electrons in their atoms. electrons assume a directed motion on the application of an electric potential difference. These electrons while flowing pass through the molecules or the atoms of the conductor, collide and other atoms and electrons, thereby producing heat. Those substances which offer relatively greater difficulty or hindrance to the passage of these electrons are said to be relatively poor conductors of electricity like bakelite, mica, glass, rubber, p.v.c. (polyvinyl chloride) and dry wood etc. Amongst good insulators can be included fibrous substances such as paper and cotton when dry, mineral oils free from acids and water, ceramics like hard porcelain and asbestos and many other plastics besides p.v.c. It is helpful to remember that electric friction is similar to friction in Mechanics.
1.5. The Unit of Resistance.
The practical unit of resistance is ohm.** A conductor is said to have a resistance of one ohm if it permits one ampere current to flow through it when one volt is impressed across its terminals. For insulators whose resistances are very high, a much bigger unit is used i.e. mega ohms =106 ohm (the prefix 'mega' or mego meaning a million) or Ohms = 103ohm (kilo means thousand). In the case of very small resistances, smaller units like milli - ohms = 10-3 ohm or micro Ohms = 10-6 ohm are used. The symbol for ohm is Q.
However. for the same resistance per unit length, cross- sectional area of aluminium conductor has to be
1.6 times that of the copper conductor but it weighs only half as much. Hence, it is used where economy
of weight is more important than economy of space.
** After George Simon Ohm (1787-1854), a German mathematician who in about 1827 formulated the law
of known after his name as Ohm's Law.
1.6 Laws of Resistance:
The resistance R offered by a conductor depends on the following factors :
(i) It varies directly as its length, I.
(ii) It varies inversely as the cross-section A of the conductor.
(iii) It depends on the nature of the material.
(iv) It also depends on the temperature of the conductor.
Neglecting the last factor for the time being, we can say that R oc ~ or R = P ~ ...(i) where p is a constant depending on the nature of the material of the conductor and is known as its specific resistance or resistivity. If in equation (i), we put I = I metre and A =I metre^22,then R =P (Fig. 1.4) Hence, specific resistance of a material may be defined as the resistance between the opposite faces of a metre cube of that material.
1.7 Unit of Resistivity.
Hence, the unit of resistivity is ohm- metre (Ohm's- m). It may, however, be noted that resistivity is sometimes expressed as so many ohm per m3. Although, it is incorrect to say so but it means the same thing as ohms- metre.
If I is in centimetres and A in cm^2,then p is in ohms- centimetre (ohm's- cm). Values of resistivity and temperature coefficients for various materials are given in Table 1.2.
Example 1.3. A coil consists of 2000 turns of copper wire having a cross- sectional area of 0.8 J mm-. The mean length per turn is 80 em and the resistivity of copper is 0.02 J1 Q~m. Find the resistance of the coil and power absorbed by the coil when connected across 110 V d.c. supply. (F.Y. Engineering. Pone Univ. May 1990) Written by Abhishek Singh E.E
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