. Field Associated with Charges: a) A static charge produces or causes an electric field around it. The strength of the field is described by a vector having direction and magnitude. b) is measured in volt (meter)^{1}or Newton (coulomb)^{1}in S.I. system. c) Direction of is radially outward for a positive ''source'' charge and radially inward for a negative ''source'' charge. d) A charge in motion produces or causes both electric field as well as magnetic field around. e) Nature of Magnetic field is described by a vector called magnetic field induction measured in tesla or weber (metres)^{2}in S.I. system.
. FORCE ON A MOVING CHARGE IN MAGNETIC FIELD a) A magnetic field interacts with the moving charges only for stationary charges its effect is zero b) Force on a moving charge in uniform magnetic field is given by , magnitude of force is F_{B}= Bqv sin q
c) When the charged particle is moving in the field direction, q = 0^{o} and F = 0 (minimum). The particle keeps moving on the same path. d) When the charged particle moves at right angles to the magnetic field, q = 90^{o} and F = Bqv (maximum) e) This force acts at right angles to both The path of the particle will be circular. f) The momentum of the particle will remain constant in magnitude but its direction will constantly be changing. g) Kinetic energy of the particle remains constant K.E = h) The magnetic force acting on the particle provides the necessary centripetal force for its circular motion. i) Radius of the circular path (here P is momentum and K is kinetic energy) j) angular velocity of the charged particle is w = k) Time period of charged particle is This does not depend on the speed of the charged particle. This depends on specific charge (q/m) or nature of particle and depends on B l) Work done by the magnetic field on the charged particle is zero.
Imp. (K and B are constant) ( B constant) (B constant)
m) when the particle enters the magnetic field at an angle with ( ^{1} 0^{o}), ^{1} 90^{o} and ^{1} 180^{o}) the path of the particle will be helical. Pitch of the helical path =
LORENTZ FORCE When a charge enters into a region where there are both electric and magnetic fields, =
. MAGNETIC FORCE ON A STRAIGHT CURRENT CARRYING CONDUCTOR a) Force on a straight conductor carrying current in magnetic field is , Magnitude of the force F = Bi / sinq
b) When B and l are parallel then = 0 and F = 0 (minimum) c) When B and l are perpendicular then = 90^{o} and F = Bil (maximum) d) Direction is at right angles to the plane containing l and B. b) Fleming's left hand rule : Stretch the forefinger, central finger and thumb of left hand mutually perpendicular. Then if the fore finger points in the direction of field () and the central finger in the direction of current (), then thumb will point in the direction of force of direction of motion.
. BiotSavart's law : a) If ab is a long wire of length 'l' then is an elemental length of it, which is given a 'vector status' whose direction is the direction of flow of current in it. b) If 'P' is a ''Field Point'' i.e a point where is being calculated due to the whole current carrying conductor then, the expressions are as follows: c) (vector form) d) (scalar form) e) is the position vector locating point 'p' ''FIELD POINT'' relative to the ''SOURCE POINT'' . f) Direction of is the direction of . g) '' is the angle between and .
Comparison of BiotSavart's law with Coulomb's law. a) Electrostatic field E Electric field is zero at infinite separation from point source charge. b) Magnetic field is zero at infinite separation from the source of elemental current. {} c) Both are long ranged (i.e) both obey inverse square law. d) In BiotSavart's law the magnetic field is produced by a vector source '''' where as in coulomb's law the electrostatic field is produced due to a scalar source Q {electric charge } e) is either parallel or anti parallel to the position vector which is drawn from source charge Q to the point 'P' where the field is being calculated, where as is perpendicular to the plane formed by and . f) Both the fields namely and obey principle of super position and this is because E B g) B in BiotSavart's law depends on the angle 'q' between and (i.e.)
where as E does not depend on the angle.
. Ampere's circuital law : ''The line integral of around any closed path equals m0I, where I is the total continuous current {steady current } passing through any surface bounded by the closed path. (mathematically) . Ampere's law describes the creation of magnetic fields by all continuous current configurations, but at our level it is useful only for calculating the magnetic field of current configurations having a high degree of symmetry { Integral should be easily evaluable }
. Magnetic Field in Some of the Well Known Cases 1) at point 'P' at a perpendicular distance 'r' from an infinitely long straight current carrying conductor is , directed into the plane of figure. 2) at point 'P' at a perpendicular distance 'r' from one end of an infinitely long straight current carrying conductor is , directed into the plane of figure. 3) due to a current carrying conductor 'ab' in the form of an arc of a circle of radius 'r' carrying current 'i'. The value of B at point 'P' is
'' is to be measured in radians. '' is the angle subtended by the arc 'ab' at the centre of the circle. If = 2pn then , directed into the plane of the figure. Where 'n' is the number of turns. 4) due to a finite straight current carrying conductor 'ab' carrying a current 'i'. The value of B at point 'P' is
* 'r' is the perpendicular distance measured from the conductor to the point 'P'. * Direction of is into the plane of the figure. 5) due to a straight finite current carrying conductor at a point 'P' which is at a perpendicular distance 'r' from one end of the conductor of length 'l' is :
from figure.
Directed into the plane of figure. 6) Two straight and infinite long parallel wires separated by a distance 'r' carry currents i_{1}and i_{2}as shown. At P, resultant induction
is { and is normally into the page }
7) At P, resultant magnetic field
8) Two straight and infinite long wires separated by a distance 'r' carry currents i_{1}and i_{2}. If P is null point where the resultant field is zero
P lies in between the wires and r_{1}, r_{2}are distances of P from the two wires
If the wires carry currents in opposite direction null point P lies externally
If i_{1}> i_{2}, then and P lies nearer to second wire bent externally. If then and P lies nearer to first wire and externally.
9) Two infinite long thin insulated straight wires carrying currents i_{1}and i_{2}lie along X and Y axes respectively as shown. Then locus of the points where the magnetic field is zero will be given by
Then locus is given y = x.
10) The figure represents a solid straight conductor carrying a current 'i' in a direction out of the plane of figure, having a circular crosssectional radius 'R'. If B_{o}, B_{x}, B_{y}and B_{z}are the magnetic field inductions calculated at points 'O', 'x', 'y' and 'z' respectively.
B_{o}= 0 {zero} , Direction as shown in figure R = oy. , Direction as shown in figure r = oz { r > R} , direction as shown in figure r = ox {r < R} Variation of B with respect to 'i' for a given 'r' Variation of B with respect to 'r', in this case for a given current 'i'. 11) A squarish wire having a total length 4L carries a current 'i' as shown in figure. The value of B at point 'P' is
Directed into the plane of the figure. 12. If current is flowing through a hollow cylindrical conductor then B = 0 in the hollow portion however B ^{1} 0 exterior to the conductor. 13. B at the centroid of an equilateral triangular shaped current carrying conductor of side length 'L' is given as : Direction of B would be into the plane if the current appears clockwise and out of the plane if the current appears counter clockwise. 14. At a point 'P' on the axis of the coil B at a point 'P' lying on the axis of a circular coil of 'n' turns each of radius 'R', carrying a current 'i' at a distance 'x' units from the centre of the circular coil is given by
Direction of will be along the axis and away from centre if current appears ''counter clockwise'' when viewed from point 'P'. At a point 'P' at the centre of the coil Put x = 0 in the above equation
At a point 'P' on the axis of the coil very far away from the centre. Put x >>>R in the above equation. ; =
Imp. Magnetic dipole moment M = i(pR^{2}) n 1) A wire of a fixed length is first shaped into circular coil of n_{1}turns and later circular coil of n_{2}turns. If same current is made to pass through then B_{1}and B_{2}are the respective magnetic field inductance at their respective centers.
2) Two circular coils are joined in series having n_{1}and n_{2}turns with radii r_{1}and r_{2}respectively. If a current 'i' is made to pass through the combination then the corresponding magnetic field inductions at the centers is B_{1}and B_{2}.
If the combination is parallel across the same cell made of same material then {independent of number of turns} have a common centre with their planes inclined at an angle ''. 3) Two circular coils, have radii r1and r2and carry currents i_{1}and i_{2}. If q is the angle between the planes of the circular loops then the resultant magnetic field inductions 'BR' at their common centre is given by
If = 90^{o} then If in addition to the above condition and then
. FIELD ON THE AXIS OF A SOLENOID a) If the point is inside the solenoid of infinite length B = b) If the solenoid is of infinite length and the point is near one end, (here 'n' is number of turns per unit length)
. FORCE BETWEEN TWO PARALLEL CURRENT CARRYING CONDUCTORS a) When the currents in both the wires are in the same direction, then mutual force between the wires is attractive. When the currents in the wires are in opposite directions, the mutual force between the wires is repulsive.
b) Mutual force between the wires is Note : 'l' is the length of the wire that experiences the force. Force per unit length on each wire is = same for both the wires in magnitude. Imp. If i = i_{2}= 1A, r = 1m then F = 2 × 10^{7}Nm^{1}. Medium being vacuum from this ampere is defined. c) The force experienced by a semi circular wire of radius 'r' when it is carrying a current 'i' and is placed in a uniform external magnetic field of induction B will be as shown.
,
Case 1 : A curved wire carries a current I and is located in a uniform magnetic field B, as shown in figure. Because the field is uniform, we can take B outside the integral in , and we obtain.
But the quantity represents the vector sum of all the length elements from a to b. From the law of vector addition, the sum equals the vector L', directed from a to b. Therefore, reduces to F_{B}= IL' × B.
Case 2 : An arbitrarily shaped closed loop carrying a current I is placed in a uniform magnetic field, as shown in fig. We can again express the force acting on the loop in the form of , but this time we must take the vector sum of the length elements ds over the entire loop : . Because the set of length elements forms a closed polygon, the vector sum must be zero. This follows from the graphical procedure for adding vectors by the polygon method. Because , we conclude that F_{B}= 0. The net magnetic force acting on any closed current loop in a uniform magnetic field is zero.
. TORQUE ON A CURRENT LOOP IN UNIFORM MAGNETIC FIELD a) When a coil current carrying is placed in a uniform external magnetic field, the net force on it is always zero. Here different parts experience forces in different directions so that the loop may experience a torque or couple. b) A rectangular coil of area A having N turns and carrying current 'i' is placed in a uniform external magnetic field of induction . If 'b' is the angle between the normal to the plane of the coil and the magnetic field, then torque acting on the coil is . In vector form Here A = lb where 'l' and 'b' denote length and breadth of the rectangle coil. If 'a' is the angle between the plane of the coil and the magnetic field, then torque acting on the coil is
If plane of the coil is parallel to the magnetic field, then the torque (maximum torque) c) When q = 0^{o} or 180^{o}, i.e. plane of the coil is perpendicular to the field, . d) Magnetic Potential energy is Defined as : . Work done in turning the coil through an angle 'q' from the field direction is W = MB (1  cos )
MOVING COIL GALVANOMETER a) In MCG, current (i) passing through the coil is directly proportional to its deflection (q) NiAB = C
K is galvanometer constant (C is restoring couple per unit twist) b) Deflection of the coil is measured using lamp and scale arrangement c) Current sensitivity of MCG is the deflection per unit current
F For more current sensitivity, B, A, N, should be more and C should be small. d) Voltage sensitivity of MCG is the deflection per unit potential difference or voltage applied across the galvanometer coil. F where G is galvanometer resistance. Voltage sensitivity also varies just as that of current sensitivity. Voltage sensitivity =
. TANGENT GALVANOMETER a) Working principle of tangent galvanometer is tangent law. b) A magnetic compass needle is placed horizontally at the centre of a vertical fixed current carrying coil whose plane is in the magnetic meridian. c) If is the magnetic induction at the centre of current carrying coil, which is perpendicular to the direction of . If the needle of magnetic compass deflects through an angle q to reach an equilibrium position, B = B_{H}tanq Here ; Here K is known as reduction factor, of the tangent Galvanometer. In this case i tanq i.e. current in the coil is directly proportional to the tangent of deflection of magnetic needle.
. SHUNT a) A low resistance connected in parallel to a galvanometer to protect it from large current is known as shunt.
USES : i) The shunt prevents the flow of large current and protects the moving coil galvanometer. Thus the life of galvanometer can be increased. ii) As it is a small resistance connected in parallel to the galvanometer, the effective resistance is very small. It is an advantage while measuring currents in a circuit. b) The main current 'i' divides into igand isas shown. Here G and S are the resistances of galvanometer and shunt respectively.
c) i_{g}G = i_{x}S as they are parallel. d) Current through galvanometer
e) Current through shunt f) Fractional current passing through the galvanometer g) Fractional current passing through the shunt h) Power of the shunt = i) If the range of the galvanometer is increased to 'n' times, then of the main current passes through the galvanometer. or
. AMMETER a) Ammeter is a device used for measuring currents in electrical circuits. b) A galvanometer can be converted into an ammeter by connecting a suitable shunt across it. c) The value of shunt resistance to be connected to convert galvanometer into ammeter is
where Here is known as sensitivity of ammeter d) Ammeter must be always connected in series, in a given circuit. e) Resistance of ideal ammeter is zero and its conductance is infinity. f) Low resistance of the galvanometer does not alter the current in the circuit. g) If the range of ammeter of resistance G_{A}is to be increased form i_{1}to i_{2}then the shunt resistance to be connected is
h) Effective resistance of ammeter when the galvanometer is shunted is
. VOLTMETER a) It is a device used for measuring potential difference between any two points in a given circuit. b) A galvanometer can be converted into a voltmeter of higher range by connecting high resistance in series. c) The value of series resistance 'R' to be connected to convert galvanometer into a voltmeter is
d) Voltmeter must be always connected in parallel in the circuit. e) Resistance of ideal voltmeter is infinity. Conductance of ideal voltmeter is zero. f) High resistance of voltmeter does not effect the potential difference measured considerably as it draws a very small current.[ potentiometer is an ideal voltmeter ] g) If voltage range of a galvanometer is increased by 'n' times by connecting a series high resistance R, then R = G(n  1) h) If the range of a voltmeter of resistance GVis increased from V_{1}to V_{2}then the resistance R to be connected in series is
R = (n  1)G (
. TOROID
Magnetic field at the centre of toroid is given as B = where N = Total no. of turns I = Current flowing in each term r = Mean radius
. CYCLOTRON
Under the action of the magnetic field, which is perpendicular to the plane of does, the ion adopts a circular path with a constant speed 'v' and of radius 'r' gives by , where B is the magnetic field induction. The time 't' required by the ion to complete a semicircle is . This shows that the time of passage of the ion through the dee is independent of the speed of the ion and of the radius of the circle, frequency u0of the applied P.D must be equal to the frequency u of the circular revolution of the ion. That is u_{o}= u therefore
LIMITATIONS : The cyclotron cannot accelerate the particles to velocities as high as comparable to the velocity of light. The reason is that at these velocities the mass 'm' of a particle increases with increasing velocity according to Kinetic Energy of Particles Accelerated in a Cyclotron : Let R be the outside radius of the dees and v_{max}the speed of the particle when traveling in a path of this radius. Then from the relation r = mv/qB, we have
The corresponding kinetic energy of the particle is
MAGNETISM . POLE STRENGTH : 1. Pole strength is a scalar and its SI units is ampere meter Dimensional formula [m] = [IL ] 2. It depends on (a) nature of the material of the magnet (b) level of magnetisation (c) directly proportional to area of cross section. 3. Pole strength is independent of the shape of the magnet.
. MAGNETIC MOMENT : 1. It is measured as the product of length and pole strength M = 21 xm. 2. Its S.I unit is ampere  metre^{2}or N  m / Tesla or N  m^{3}/ weber [M] = [IL^{2}]. 3. It is a vector with its direction from south pole to north pole along its axial line . 4. M = I × V or M = I Intensity of magnetisation V magnetised volume, C moment of the couple. 5. If a magnet is cut into n equal parts, in a direction parallel to its length, then a) Pole Strength of each part m1= b) Length of each part, 2l1= 2l c) Magnetic moment of each part, 6. When a bar magnet is cut into 'n' equal parts normal to its axis. a) Pole strength of each part m1= m. b) Magnetic moment of each part M1= c) Length of each part 211= . 7. When a bar magnet is cut into 'xy' equal parts x parts parallel to its axis and y parts normal to the axis. a) pole strength of each part m1= b) Magnetic moment of each part M1= c) Length of each part 211= 8. When a thin bar magnet of magnetic moment 'M' and length '2l' is bent at its mid point with an angle '' between the two parts, its new magnetic moment M1= M sin (/2). 9. When a magnetised wire of length '2l' and magnetic moment 'M' is bent into an arc of a circle, that makes an angle '' at the centre of the circle. a) Its length decreases and becomes b) Its pole strength remains same as 'm' c) Its magnetic moment decreases becomes M1= where is in radian d) Magnetic moment of magnet making an angle at the centre of circle is M. If it is made straight, it's magnetic moment becomes 10. When two magnets of magnetic moments M_{1}and M_{2} kept at an angle '' with like poles touching each other, then the resultant magnetic moment M^{1}= 11. When two magnets of magnetic moments M_{1}and M_{2}are kept at an angle '' with unlike poles touching each other, then the resultant magnetic moment M^{1}= 12. When 'n' identical bar magnets form a closed polygon with unlike pole nearer the resultant magnetic moment is zero. 13. If one magnet is removed from the polygon the resultant magnetic moment becomes equal to the magnetic moment of one magnet (M). 14. If one magnet in the polygon is reversed and kept at the same position the resultant magnetic moment becomes '2M'. 15. If small holes are made in the body of the magnet, its magnetic moment decreases because its magnetised volume decreases.
. COULOMB'S INVERSE SQUARE LAW : The force between two isolated magnetic poles is expressed by a) F = b) F = . If poles are in air and medium is between the poles. c) F = . If poles and space between the poles are in medium. d) _{o} = 4 × 10^{7}henry / metre or or . e) _{r} = Relative permeability of the medium = = . f)_{r} = 1 for air or vacuum. g) _{r} = h) = 10^{7}henry / metre is the rationalised constant in S.I system.
. MAGNETIC INDUCTION OR INDUCTION FIELD STRENGTH OR MAGNETIC FLUX DENSITY : 1. It is numerically equal to the force experienced by unit north pole kept at the given point a) 'B' is a vector having direction in the direction of auxiliary field H. b) Dimensional formula [B] = = [MT^{2}I^{1}]. c) S.I unit weber / m^{2}or N/A  m or tesla d) 1 tesla = 10^{4}gauss e) B depends on the medium 2. Magnetic induction due to an isolated pole of strength 'm' is given by B = in air, away from pole of it is Npole & towards the pole if it is Spole 3. Magnetic flux per unit area is called magnetic induction of flux density B = . 4. Magnetic flux density or induction at any point over the surface of a sphere of radius 'r' which is having a north pole of strength 'm' ampere metre at its centre is given by B = = = 5. When a pole of pole strength 'm' is placed in a magnetic field of induction B, the force F experienced by it is given by F = mB 6. North pole experiences the force in the direction of field. South pole experiences the force in the direction opposite to that of the field.
. INTENSITY OF THE MAGNETIC FIELD OR MAGNETISING FIELD STRENGTH (H) : a) 'H' is an auxiliary field which is measured as the ratio of magnetic induction to the permeability of the medium at the given point b) H = (in medium) and H = in air or vacuum c) H is a vector in the direction of the force. d) I.M.F = [H] = [IL^{1}] e) S.I unit of H is Am^{1} f) 1 Am^{1}= 4 × 10^{3}oersted g) H is independent of the medium h) Intensity of the magnetic field due to an isolated pole of strength m is given by H = = (in air or medium)
. MAGNETIC FLUX : 1. The total number of magnetic lines of induction passing normal to the given cross section is called magnetic flux. It is measured as the dot product of magnetic induction and areal vector . = BA cos q is the angle made by magnetic induction with the normal to the area a) It is a scalar b) S.I unit of flux weber c) 1 weber = 10^{8}maxwell d) Dimensional formula [f] = = = [ML^{2}T^{2}I^{1}] 2. Magnetic flux due to an isolated pole of strength 'm' is = _{o} m (in air) and = m = _{o}_{r}m (in medium) 3. Magnetic flux due to a magnet over a closed surface is zero = = 0 (Gauss' theorem in magnetism) 4. Force experienced by magnetic pole of strength 'm' in a uniform magnetic field 'B' is given by F = mB. a) North pole experiences force in the direction of the field. b) South pole experiences force opposite to the direction of the field.
. MAGNETIC LINES OF FORCE : 1. Magnetic field can be represented by magnetic lines of force. 2. A magnetic line of force is the path followed by a free unit north pole in a magnetic field. 3. Two magnetic lines of force never intersect, if they intersect it means that field has two directions at the point of intersection which is impossible. 4. They diverge from north pole and converge at south pole. Inside the magnet they travel from south pole to north pole. outside the magnet they travel from north pole to south pole. They are closed curves. 5. Magnetic lines of force pass through the magnetic substance of the magnet. Electric lines do not pass through a conductor. 6. They contract longitudinally and expand laterally. 7. The tangent at any point to the line of force gives the direction of at the point. 8. In a uniform magnetic field lines of force are equidistant parallel straight lines. 9. In non uniform magnetic field lines of force are curves. 10. At null points resultant = 0 and hence no lines of force pass through that point. 11. If lines of force are crowded it implies stronger field. 12. Magnetic lines of force eminate or end on the surface of the magnet at any angle. Electric lines of force eminate or end always normally on the surface of the conductor.
. TORQUE ACTING ON A BAR MAGNET IN A UNIFORM MAGNETIC FIELD. 1. When a magnet with magnetic moment M is suspended in a uniform field of induction B at an angle q with the field direction then the couple acting on the magnet. C = MB sin q and vectorially 2. When = 90^{o}is maximum C_{max}= MB If = 900and B = 1, C_{max}= M 3. When = 0^{o} C = 0 4. In a uniform magnetic field a bar magnet experiences only a couple but no net force. Therefore it has only rotatory motion. 5. In a non  uniform magnetic field a bar magnet experiences a couple and also a net force. So it undergoes both rotational and translational motion. 6. A bar magnet of moment M is initially parallel to the magnetic field of induction B. The angle through which it must be rotated so that it experiences half of the maximum couple is 30^{o}. C = = MB sin q = = 30^{o} 7 A bar magnet of moment M is initially perpendicular to the magnetic field of induction B. The angle through which it must be rotated so that it experiences half of the maximum couple is 600. 8. The work done in deflecting a magnet from angular positions q1 to an angular position q2 is W = C q = MB sin q W = MB (cos _{1} cos _{2} ) 9. The work done in deflecting a bar magnet through an angle from equilibrium position in a uniform magnetic field is given by W = MB (1  cos ). If = 1800 W = 2MB (Maximum) 10. P.E of the magnet =  =  MB Cos where = 0^{o}PE =  =  MB (P.E Minimum stable equilibrium) = 90^{o}P.E =  = 0 = 180^{o}P.E =  = + MB (P.E Maximum unstable equilibrium) 11. Two magnets of magnetic moments M_{1}and M_{2}of equal mass are joined in the form of a cross and this arrangement is pivoted so that it is free to rotate in a horizontal plane under the influence of earth's magnetic field. If is the angle made by the magnetic meridian with M1then tan = 12. A bar magnet is under the influence of two magnetic fields At equilibrium C_{1}= C_{2} MB_{1}sin = MB_{2}sin =
13. A thin bar magnet is pivoted at its centre. It is held in equilibrium at an angle to a magnetic field B by applying a force F at a distance r from the pivot along a direction making an angle with, then = C r F sin = MB sin
14. A magnet is suspended in magnetic meridian with untwisted wire. The upper end of wire is rotated through a to deflect the magnet by q from magnetic meridian then C = MB_{H}sin 15. Force between two coaxial magnetic dipoles F = 16. The force between two magnets which are perpendicular to each other is F =
. FIELD ON AXIAL LINE : 1. B at any point on the axial line of a bar magnet is B = 2. In case of short bar magnet l < < d B = 3. Direction to is form S to N along the axial line i.e, along 4. B when M is constant = 5. Two magnets of different lengths having same moment are taken. Magnetic field at distance 'd' from the centre on the axial line is more of the longer magnet.
6. H = 7. Force experienced by a pole of pole strength 'm' placed at a point on the axial line of a short magnet F =
. FIELD ON EQUITORIAL LINE : 1. B at any point on the equatorial line of a bar magnet is B = 2. In case of short magnet B = 3. B B_{1}= B_{2} 4. Direction of is parallel to the axis and is directed from N to S pole i.e . 5. Two magnets of different lengths having same M are taken. B at distance 'd' from the centre on the equatorial line is more for shorter magnet 6. The ratio of fields at a distance '' on the axial and equatorial lines of a short bar magnet is 7. B at any point P B =
. NEUTRAL POINTS OR NULL POINTS : 1. It is the point in a combined magnetic field where the resultant magnetic field is zero. 2. No line of force passes through the null points. 3. A compass needle placed at null point comes to rest in any direction. 4. If two poles of pole strength m_{1}and m_{2}are separated by distance d, then the distance of the neutral point from the first pole m1is X = For like poles the neutral point is situated in between the poles. For unlike poles the neutral point is situated outside the line joining the poles. 5. If two short bar magnets of magnetic moments M_{1}and M_{2}are placed along the same line and 'd' is the distance between their centers, the distance of null point from M1is X = '+' when like poles are facing each other when unlike poles face each other null point is not possible. 6. When a short magnet is placed in the earth's magnetic field with its north pole towards geographic north. i) null points are formed on the equatorial line. ii) at the null point =  BH= 3. 768 × 10^{5} is the horizontal component of earth's magnetic field. iii) B_{H}= 7. When a short magnet is placed in earth's field with south pole pointing geographic north i) null points are formed on axial line ii) B_{H}= 8. If a very long magnet is placed vertically with one pole on the table i) a single neutral point will be formed ii) If 'm' is the pole strength and 'd' is the distance from the pole of the magnet where the neutral point is formed then B_{H}= iii) If the north pole is on the table, then the neutral point lies towards geographic south. iv) If the south pole is on the table, then the neutral point lies towards geographic north.
. DEFLECTION MAGNETOMETER : 1. Deflection magnetometer works on the principle of tangent law 2. Tangent law B = B_{H}tan B = Magnetic field due to bar magnet. = horizontal component of earth's magnetic field. = the angle through the needle deflects. 3. Tan  A position or Gauss  A position or End on position. i) Arms of the magnetometer are perpendicular to the magnetic meridian i.e. arms lie in EastWest direction. ii) Aluminium pointer is parallel to the arm iii) The bar magnet is placed in such a way that its axial line passes through the centre of the needle. Bar magnet is parallel to the arms. iv) = B_{H}tan (For short magnet) = _{H}Htan ( For long magnet)
4. Tan  B position or Gauss B position or broad side on position i) arms are in the magnetic meridian i.e, in the north south direction. ii) Aluminium pointer is perpendicular to the arms. iii) The bar magnet is placed in such a way that its equatorial line passes through the centre of the needle. The magnet is perpendicular to the arms. iv) (for short magnet)
(for long magnet) 5. The Aluminium pointer in the compass box provides stability to the needle and enables us to have a large circular scale. 6. The plane mirror is provided below the Aluminium pointer to note the deflections without parallax error. 7. Both in tan A as well as tan B position the bar magnet is in eastwest direction. 8. Tan A position is preferred to Tan  B position because for the same distance between the centre of the needle and deflection magnetometer deflection is more in TanA position ( the field is more on axial line than equatorial line) 9. Deflection magnetometer is used to i) measure the magnetic field ii) compare the magnetic moments of two bar magnets. iii) compare the magnetic field strengths iv) Verify the inverse square law. 10. Equal distance method : For Tan A or Tan B position = . 11. Null Deflection Method: For Tan A or Tan B position = null deflection is preferred to equal distance method. 12. In tan A position, null deflection method the magnets are placed with their like poles facing each other. 13. In tan B position, null deflection method the bar magnets are placed such that their unlike poles lie on the same side of the deflection magnetometer. 14. When a short magnet is placed at a distance '' in tan A and then tan B position and _{A} and _{B} are the average deflection then =2(verification of inverse square law) 15. Only one bar magnet is used while verifying inverse square law. 16. In single pole method or Robinson's method, the inverse square law is verified if = B_{H}tan or ^{2}tan = const. 17. Deflection magnotometer does not work at magnetic poles of the earth because B_{H}= 0 18. If magnets of magnetic moments M_{1}and M_{2}are placed one on the other with like poles touching each other on the arm of a magnetometer, the average deflection is _{1}, when one of the bar magnets is reversed the average deflection is _{2}, Then = 19. a) Deflection magnometer is more accurate when the deflection is 45^{o}. Therefore the reading should lie between 30^{o}to 60^{o}for accurate results. b) The relative error is minimum when = 45^{o}. 20. Earth's horizontal component of magnetic field B_{H}at two places can be compared using a single magnet at same distance from the centre of the needle in the two places = 21. Using deflection magnetometer is determined
. VIBRATION MAGNETOMETER : 1. It works on the principle that when a bar magnet is suspended freely in a uniform magnetic field and is displaced from its equilibrium, it starts vibrating and executes SHM about the equilibrium position. 2. The plane mirror with the index line helps to count the vibrations of the magnet and avoids the error due to parallax. 3. The time period of vibrating T = where is the moment of inertia of the bar magnet about the axis of oscillation. m = mass of the magnet l = length of the magnet b = breadth of the magnet B_{H}= horizontal component of earth's magnetic field 4. frequency of oscillation (n) = 5. A magnet is oscillating in a magnetic field and its time period is T. If another identical magnet is placed over that magnet with similar poles together, then time period remains unchanged I^{1}= 2I, M^{1}= 2M, T1= = = T 6. A magnet is oscillating in a magnetic field B and its time period is T. If another identical magnet is placed over that magnet with like poles together, then time period becomes infinite
7. Two magnets of magnetic moments M_{1}and M_{2}(M_{1}> M_{2}) are placed one over the other. If T_{1}is time period when like poles touch each other and T_{2}is the time period when unlike poles touch each other, then in sum and difference method = . 8. If n1_{1}and n_{2}are frequencies when like poles touch each other and unlike poles touch each other then. 9. Using the same bar magnet in deflection magnetometer in tan A position and vibration magnetometer.
M = = deflection in deflection magnetometer I = Moment of inertia of bar magnet 2l = Length of magnet T = Time period of vibration d = distance between magnet and centre of needle in deflection magnetometer
. Elements of Earth's Magnetism The physical quantities which completely describe earths magnetic field are called elements of earth's magnetism. These are declination, dip or inclination and horizontal component of earth's magnetic field. Declination (q) : Declination at a place is the angle between the geographic meridian and the magnetic meridian at that place. It is denoted by q. (i) (ii) The angle of declination varies from place to place on the earth's surface. The declination is helpful is steering ships in the right direction with the help of mariner's compass. Dip or Inclination: Dip or inclination at a place is the angle between the direction of intensity of earth's total magnetic field (R) and the horizontal direction in the magnetic meridian at that place. It is denoted by d. At the earth's magnetic poles, the magnetic field of earth is perpendicular to earth's surface. Therefore, the value of dip is 90^{o} at earth's magnetic poles: The dip needle becomes vertical at these locations. At the magnetic equator, d = 0^{o} so that dip needle becomes horizontal.
Horizontal component of Earth's magnetic field: It is the component of earth's total magnetic field along horizontal direction in the magnetic meridian. It is denoted by H. Thus the referring to figure (i), R is the intensity of the earth's total magnetic field and d is the angle of dip. Resolving R into two rectangular components, we have, Horizontal component, H = R cos ... along AP Vertical component, V = R sin ... along AQ R^{2} = H^{2}+ V^{2} tan = Note. Once we know the values of declination (q), dip (d) and horizontal component (H) of earth's magnetic field at a place, we can specify the strength and direction of earth's magnetic held at that place.
Useful Information About Dip H = R cos and V = R sin (i) At a place on the poles, = 90^{o}. V = R and H = 0 Therefore, earth always has a horizontal component except at poles. The dip needle becomes vertical at poles. (ii) At a place on the equator, = 0^{o} V = 0 & H = R Therefore, earth always has a vertical component except at the equator. The dip needle becomes horizontal at the equator. (iii) If a dip needle is suspended in a plane at an angle of q^{o} with the magnetic meridian and apparent dip is d' in this plane, then tan = tan ' cos (iv) If _{1}and _{2} are observed angles of dip in two arbitrary vertical planes perpendicular to each other, then true dip d is given by cot^{2} = cot^{2} _{1}+ cot^{2} _{2}.
. Magnetic Maps The value of all the three magnetic element, i.e., declination, dip and horizontal component are found to be different at different places on the surface of earth. Their values are determined all over the globe and the results are represented on geographical maps. Usually lines are drawn joining all places having same value of an element. Such maps are called 'magnetic maps'. However, earth's magnetic field at a given place is found to change with time, so magnetic maps are revised from time to time. Following are the important lines on a magnetic map. Isogonic lines: Lines drawn through different places having the same declination are called isogonic lines. The line which passes through placed having zero declination is called agonic line and along it the compass needle will point geographical north. So a place east of agonic line will have west declination as the compass needle points to west of north. Isoclinic lines: These are lines passing through places of equal dip. The line joining places of zero dip is called aclinic line (or magnetic equator). At all places upon this line a freely suspended magnet will remain horizontal. To the north of this line the north pole of the needle dip downwards while to the south of it, reverse will occur.
Tangent Law in Magnetism According to tangent law, if two uniform perpendicular magnetic fields V and H act simultaneously on a magnet, the magnet comes to rest making an angle q with the direction of H such that:
tan = V = H tan
. Magnetic Properties of Matter Atom As A Magnetic Dipole We now derive an expression for the magnetic dipole moment due to orbital motion of electrons figure shows an electron revolving in an orbit of radius r with an angular velocity w. The circulating electron is equivalent to a singleturn current loop. The magnetic dipole moment of this current loop is given by; M = iA ––––––––––– Now
M = iA = ...(i) According to Bohr's theory, the angular momentum of an electron (mvr) in a stationary orbit can have only those values which are integral multiples in h/2p, i.e. ..(ii) where n = 1, 2, 3, .... From (i) and (ii)
or The least value of the magnetic dipole moment of an electron due to orbital motion occurs when n = 1. This is called Bohr magneton and is represented by mb. M = n _{B} where Definition of B: The magnetic moments of atom are expressed in terms of _{B}.
A Bohr magneton is equal to the orbital magnetic moment of an electron circulating in an orbit with the smallest allowed value of orbital angular momentum The magnetic dipole moment of atom is of the order of a few Bohr magnetons.
Important Terms: Consider an ironcored toroid carrying a current I and having n turns per unit length if the absolute permeability of iron is (= 0r), then total magnetic flux density (B) in this material is B = n I ...(1) Magnetising force or magnetic intensity The quantity n I in equation (1) is called the magnetising force or magnetic intensity i.e. H = nI. B = H Hence, magnetising force may be defined as the number of ampereturns flowing per metre length of the toroid. Clearly the unit of H is ampere per metre (A/m). Thus if a toroid has 10 turns per metre length and current flowing is 2A, then magnetising force is H = nI = 10 × 2 = 20 A/m. (i) The ratio B/H in a material is always constant and is equal to the absolute permeability (= 0r) of the material. (ii) If the same magnetising force H is applied to two identical aircored and ironcored toroids, then magnetic flux density produced inside the toroids is B_{0}= _{0}H ...air/vacuum B = _{0}_{r}H ... in iron, where _{r}is the relative permeability of iron.
Intensity of magnetisation When a magnetic material is subjected to a magnetising force, the material is magnetised. Intensity of magnetisation is a measure of the extent of which the material is magnetised and depends upon the nature of the material. It is defined as under: The intensity of magnetisation of a magnetic material is defined as the magnetic moment developed per unit volume of the material. Intensity of magnetization , where = magnetic volume developed in the material V = volume of the material. If m is the pole strength developed, a is the area of crosssection of the material and 2l is the magnetic length, then, I = Hence intensity of magnetisation of a material may be defined as the pole strength developed per unit area of crosssection of the material.
Comparison Chart of Diamagnetic, Para and Ferromagnetism Sr. No.  Diamagnetism  Paramagnetism  Ferromagnetism  1.  Substances are feebly repelled by the magnet.  Substance are feebly attracted by the magnet.  Substances are strongly attracted by the magnet.  2.  Magnetisation I is small, negative, and varies linearly with field.  I is small, positive and varies linearly with field.  I is very large positive and varies nonlinearly with field.  3.  Susceptibility cmis small negative and temperature independent.  cmis small, positive and varies inversely with temperature i.e. cmµ (1/T).  cmis very large, positive and temperature dependent.  4.  Relative permeability µris slightly lesser than unity i.e., µ < µ0  µris slightly greater than unity i.e., µ > µ0  µris much greater than unity i.e., µ >> µ0.  5.  Lines of force are expelled from the substance, i.e B < B0.  Lines of force are ‘pulled in’ by the substance, i.e., B > B0.  Lines of force are ‘pulled in’ strongly by the substance i.e. B >> B0.  6.  It is practically independent of temperature.  It decreases with rise in temperature.  It decreases with rise in temperature and above Curie temperature becomes paramagnetic  7.  Atoms do not have any permanent dipole moment.  Atoms have permanent dipole moments which are randomly oriented.  Atoms have permanent dipole moments which are organised in domains.  8.  Exhibited by solids, liquids and gases.  Exhibited by solids, liquids and gases.  Exhibited by solids only that too crystalline.  9.  Bi, Cu, Ag, Hg, Pb, water, hydrogen, He, Ne, etc. are diamagnetic.  Na, K, Mg, Mn, Al, Cr, Sn and liquid oxygen are paramagnetic.  Fe, Co, Ni and their alloys are ferromagnetic.  Diamagnetic Element
Paramagnetic Element
Ferromagnetic Element
Curie Law
According to Curie law, for away from saturation the magnetic succeptibility of a paramagnetic substance is inversely proportional to absolute temperature.
...(1) when a ferromagnetic material is heated, it becomes paramagnetic at a certain temperature. This temperature is called curie point or curie temperature (T _{c} ). For ferromagnetic substance we have, C is called Curie constant.
T _{c} = Curie point T (> T _{c} ) = temperature above curie point. C' = Another Curie Constant The Curie point of iron is 1043 K.
Hysteresis
When a ferromagnetic substance (e.g. iron) is subjected to a cycle of magnetisation (i.e. it is magnetised first in one direction and then in the other), it is found that intensity of magnetisation (I) in the material lags behind the applied magnetising force H. This phenomenon is known as hysteresis. The phenomenon of lagging of the Intensity of magnetisation (I) behind the magnetising force (H) in a ferromagnetic material subjected to cycles of magnetisation is known as hysteresis.
The term hysteresis is derived from the Greek word hysteresis meaning to lag behind. If a piece of ferromagnetic material is subjected to one cycle to magnetisation, the resultant (I)  H curve is a closed loop a b c d e f a called hysteresis loop see figure (ii). Note that (I) always lags behind H. Thus, at point b, H is zero but I has a finite positive value ob. Similarly, at point e, H is zero but Intensity of magnetisation (I) B has a finite negative value one. Retentivity : The value of I retained by the material when magnetising force H is zero, is called retentivity. In the diagram, value of ob is retentivity Coercive Force (HC)
The value of magnetising force in direction reaction to make residual magnetism zero is called coercive force (H C ).
Hysteresis Loss
When a ferromagnetic material is subjected to cycles of magnetisation, the domains of the material resist being turned first in one direction and then in the other direction. Energy is thus expended in the material to overcome this opposition. This loss is in the form of heat and is called hysteresis loss. The area of hysteresis loop is proportional to the thermal energy developed per unit volume of the material as it goes through the hysteresis cycle. Hysteresis loss is present in all those electrical machines whose iron parts are subjected to cycles of magnetisation. The obvious effect of hysteresis loss is the rise of temperature of the machine.
Importance of Hysteresis Loop
The shape and size of hysteresis loop largely depends upon the nature of the material. The choice of a ferromagnetic material for a particular application then depends upon the shape and size of the hysteresis loop. Properties of material of electro magnet
(1) High value of saturation magnetisation. (2) Low coercivityb (3) Low retentivity (4) Small area of hysteresis loop. Far example soft iron.
Properties of material of permanent magnet
(1) High value of saturation magnetisation. (2) Large coercivity (3) Large retentivity For example: Steel
