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ELECTROMAGNETIC UNITS AND DIMENSIONS THE S.I. SYSTFM OF UNITS We shall employ only one system of electrical units in this section. This is the "rationalised M.K.S. or S.I. system", based on the metre, the kilogram and the second as the fundamental units, and on the ampere as the basic electrical unit. Electric quantities are measured in terms of the coulomb, the volt, the ohm and other “practical” units and mechanical quantities in terms of the metre, the kilogram and the second. The unit of force is the newton. We define the newton as that force which, when acting on a mass of 1 kg, imparts to it an acceleration of 1 metre per second. Since F = ma, 1 newton = 1 kg m sec-2 = 105 dynes We next define the unit of work or energy, i.e. the joule, as the work done by a force of 1 newton acting through a distance of 1 metre. Since W = Fd, 1 joule = 1 newton metre = 107 dyne cm (ergs) The watt becomes the unit of power, being defined as the rate of working of 1 joule per second. The unit of current is the ampere, the definition of which is based on the force between two parallel, very long and straight wires carrying currents. Experiments show that the force per unit length of the conductor depends on the currents and the distance between the wires according to the equation dF II k 1 2 d r 35 By giving a value of 2 x 10-7 to k, the factor of proportionality, the ampere is defined. The ampere is then that unvarying current which, flowing in each of two parallel, very long and straight conductors of negligible cross section placed in vacuum at a distance of 1 metre apart, produces between these conductors a force of 2 x 10-7 newtons per metre of their length. For convenience, k is put equal to 0 /2 so that : 0 k 2 10 7 N A 2 2 or 0 4 10 7 N A 2 where 0 is known as the Permeability of Free Space. We obtain immediately the unit of electric charge, the coulomb as the charge transferred when a current of 1 ampere flows for 1 second. ELECTROSTATICS Measuring charges in coulombs, the force between two charges in vacuum becomes q1 q 2 1 q1 q 2 F 8.987 10 9 r 2 4 0 r 2 so that 0 8.85 10 12 C 2 N m 2 0 is known as the Permittivity of Free Space. It is found to be related to 0 by the equation c 2 1 0 0 where c is the velocity of light (3 x 108 m/sec). 36 The units of 0 and 0 are usually expressed in henrys per metre and farads per metre respectively, i.e. 0 4 10 7 H m or N A 2 C2 0 8.85 10 12 F m or Nm 2 The potential difference between two points represents the work done per unit charge in transferring the charge from one point to the other. We therefore express the unit of potential difference in joules per coulomb, otherwise known as the volt. An electric field exists at a point in space if a charged conductor placed at the point experiences a force. The electric field intensity at a point in space is defined as the force per unit charge acting on the charge placed at that point. F E q The unit of Electric Intensity is therefore the newton per coulomb. The work done per unit charge in moving a charge through a distance d against the force is dV = -E d so that dV E d From this equation, the unit for measuring electric intensity is the volt per metre. The newton per coulomb and the volt per metre are equivalent. The expression for the field due to a charge q at a distant point becomes 1 q E V m 4 0 r2 37 1 q and the potential V volts 4 0 r We define the Electric Displacement or Electric Induction D by the equation D 0 E for free space. C2 N The unit of displacement is 2 i.e. the coloumb per metre2 . Nm C The unit of capacity of a conductor or condenser is the coloumb per volt, known as the farad. The resistance of a conductor is defined from Ohm's law. The unit is the ohm - the resistance of a conductor which carries a current of 1 ampere with a potential difference of 1 volt applied to its ends. The unit of resistivity is the ohm-metre. ELECTROMAGNETISM When a current flows in a conductor placed in the neighbourhood of another conductor also carrying a current, a force is found to be exerted between them. Similarly a charge moving in the vicinity of another moving charge is found to experience a force (over and above the electrostatic force). A magnetic field may be conceived to be established by one current (or moving charge) and this field then acts on the second current (or moving charge) in the field. A magnetic induction or field B is said to exist at a point in space if a conductor carrying a current placed at the point experiences a force. Experiments show that the force depends on the strength of the current, the length of the conductor and on the direction of the current. (Note that B is a vector quantity). We define the direction of the field as that orientation of the current that experiences zero force, and the magnitude of the magnetic induction B at a point as the force per unit length acting on a conductor carrying a current of 1 ampere placed 38 normal to the field. In the special case of a straight wire length , placed normal to a homogeneous magnetic field B, the force F = I B The direction of F being given by the familiar left-hand rule. We can see from the above equation that the induction B has a dimension of newton per ampere metre. A coil of area A and N turns oriented with its plane parallel to a uniform field B will therefore experience a torque given by NIAB newton metre. The magnetic flux through a surface A is defined as the product of the area of the surface A and the component of the induction B normal to the surface. = BA cos where is the angle between vector B and the normal to the surface. Experiments show that a changing flux through a circuit induces an e.m.f. equal to the negative rate of change of flux through the circuit (Faraday's law). d i.e. dt The equation defines the unit of flux, known as the weber. The weber is that change in flux through a circuit which taking place in 1 second produces an e.m.f. of 1 volt in the circuit. The equation shows that the weber has the dimension of volt second. Since = BA, the unit of magnetic induction may be expressed in weber per square metre. The unit of B is expressed more commonly in weber/square metre than in newton per ampere metre. The unit is also called a Tesla. 39 N N m sec V sec Wb Note 2 Am C m2 m2 m The magnetic induction B is (from the equation B = /A) a flux density. A useful concept pertaining to the magnetic field B is the concept of lines of magnetic induction or flux lines. These lines are imagined to fill the space occupied by the field such that the direction of a line at any point is the direction of the vector B at the point and such that the concentration of the lines at the point (i.e. the number of lines per unit area normal to the lines) is set equal to the magnitude of B at the point. number of induction lines i.e. B normal area A line of magnetic induction has the dimension of B x area and is therefore equal to one weber. The magnetic field B which acts on a current or moving charge is set up by some other current or moving charge. For instance when a current I flows in a long solenoid, a uniform magnetic field is produced inside the solenoid. This magnetic field is found to be dependent on the medium in the solenoid, the current I and the number of turns per metre N NI B NI where is a factor dependent on the medium. may be regarded as the cause which NI results in the magnetic flux inside the solenoid. is termed the magnetising force H of the magnetic field (also known as the magnetic field intensity or magnetic field strength). H is a vector, usually having the same direction as the field B. 40 Since H N I , the unit of H is expressed in ampere-turn per metre (AT/m). The ratio of the magnetic induction B to the magnetising force H is called the absolute permeability or simply the permeability of the medium. B = H For free space 0 Wb m Wb has dimensions : 2 m A Am The weber per ampere metre has the same dimension as henry per metre (H/m). The magnetising force for other circuits may be derived from Biot and Savart's law: 1 I d sin H Am 4 r2 where r is the distance of a point in the magnetic field from an element d of the circuit carrying current I, and the angle between d and r. Integration of this equation gives : I (a) for a long straight wire: H Am 2r 0 I B Wb m 2 2r where r is the distance of the point from the wire. I (b) for a circular coil: H Am 2a 0 I B Wb m 2 2a 41 where B is the field at the centre of the coil of radius a. At a great distance r along the axis of the coil: I a2 I a2 H 2(a 2 r 2 ) 3 2 2r 3 1 2M Putting M = Ia2 H Am 4 r 3 0 2M B Wb m 2 4 r 3 M is termed the Magnetic Moment of the coil and is defined by M = IA where A is the area of the coil. The dimension of M is Am2. 0 I1 A very long wire carrying a current I1 produces a field B1 at a point 2r distant r from it. It will therefore exert a force on another wire length carrying a current I2 given by: 0 I1 F I 2 B1 I 2 2r F I I The force per unit length 0 1 2 2 r We employ this equation to define the unit of current, the ampere, by giving an arbitrary value of 4 x 10-7 to 0 . A coil carrying a current I1 produces a magnetic field. If another coil of N2 turns is placed in the vicinity of the first coil, a certain amount of the flux will pass through the second coil, each line of induction linking with N2 turns. The flux linkage (N2) linked with the second coil depends not only on the current I1 but also on the 42 relative position and geometry of the coils. The relation between and I1 may be written : N 2 M 12 I1 . M12, the flux linked with one coil when unit current flows in the other, is known as the Mutual Inductance of the circuits. Further, if the current I1 changes, the flux linkage N2 also changes and consequently an emf 2 induced in the second coil. d dI 2 ( N 2 ) M 12 dt dt assuming M12 is independent of the current. This equation is also used to define M12. The unit of mutual inductance is the henry. It follows from the definition the henry has the dimension: sec ond Wb henry volt Ampere A A coil in which the current is changing has its changing flux linked with itself so that a back emf is self induced in the coil: dI L dt L is known as the Self Inductance. The self inductance of a long solenoid of length and cross section A in air is 0 N 2 A L henries. If a secondary coil of N2 turns is wound round the middle of a long solenoid of length 1 , cross section A1 and turns N1, the mutual inductance: 43 0 N1 N 2 A1 M henries 1 MAGNETISM The magnetic induction or flux density B at the centre of a long solenoid or of a toroidal coil is increased greatly if the core is iron instead of air. The increase in flux is due to the revolving electronic charges or their spin about some axes each of which is equivalent to a current i. If a is the area of the orbit, each atom has then a magnetic dipole moment ia (Wb - m). The magnetising force H = NI/ of the solenoid orientates these electronic currents so that there is a resultant moment in the direction of the magnetising force. The resultant magnetic moment per unit volume of the material is known as the Intensity of Magnetisation or Magnetisation J. J = nia where n is the effective number of electronic currents per unit volume in the direction of N. It is a vector having the same direction as H. The intensity of magnetisation is the flux density contributed by the electronic current. The resultant magnetic induction can therefore be written B = 0 H + J J has the same unit as B i.e. the weber per square metre. Since the magnetisation is brought about by the magnetising force H, a relation exists between J and H for the material. The ratio of the magnetisation to the magnetising force is known as the Magnetic Susceptibility of the material: J = H. We may write B 0 H J ( 0 ) H H 44 When the magnetising current of a solenoid having a ferro-magnetic core is removed, the core is found to retain some of the magnetisation, i.e. the electric currents remain oriented to some extent so as to give a resultant magnetic moment. The material is said to be a permanent magnet. The flux from the electronic currents extend to the surrounding medium, the flux lines in the medium appearing to originate from the ends of the magnet. Magnetic charges called Poles - North and South Poles - may be imagined to reside at the ends of the magnets. This concepts of fictitious magnetic charges is useful for computation purposes. If the magnetic moment of a magnet of length and cross section A is M, the Pole Strength m is defined by M = m The unit pole consequently is the weber. The pole strength is related to the electronic currents: M JV niaA m m niaA webers A magnet placed with its length perpendicular to a field B will, like a solenoid of the same moment, experience a torque : = MB (Nm). The same result is obtained if a force mB is thought to act on each of the two poles, i.e. a pole m in a field B experiences a force given by F = mB newtons If a single N-pole m is imagined to be placed at the centre of a coil of radius r carrying a current I, it may be assumed that a field B is produced at the coil radially from m. A force 2rIB acts on the coil in the direction normal to the plane of the coil. 45 0 I The force on the pole m in the field of the coil is m . These forces must be equal 2r and opposite. Hence 0 m B Wb m 2 4 r 2 This shows that the field due to m follows the inverse square law. The force between two poles is then: 0 m1m2 F mB 4 r 2 46