In this paper we are elaborate Electrostatics is based primarily inverse law. Then small volume distance between two forces. Also we solve the Uniqueness of the solution of an electrostatic problem 

P, ELAVARASI

English

In this paper we are elaborate Electrostatics is based primarily inverse law. Then small volume distance between two forces. Also we solve the Uniqueness of the solution of an electrostatic proble

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A  STUDY ON   ELECTRO STATICS (P.ELAVARASI,RRAMYA,M.MAHALAKSHMI,Dr.S.SANGEETHA) (ramya25071987@gmail.com,sangeethasankar2016@gmailcom) Department of Mathematics Dhanalakshmi Srinivasan College of  Arts and Science for Women (Autonomous) Perambalur ABSTRACT:                                   In  this  paper we are elaborate Electrostatics is based primarily inverse law. Then small volume distance between two forces.  Also  we  solve the Uniqueness of the solution of an electrostatic problem .  Keywords:   Electro , vector , volume , surface  , direction ,magnitude Introduction: Vector calculus, or vector analysis is a branch of mathematics concerned with differentiation and integration of vector fields. “Vector calculus” is some times used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Electrostatics is concerned with electricity at rest. This electricity may be in the form of discrete point charges or in the form of a continuous charge distribution with volume density   or surface density   or even in the form of electric dipoles, where a dipole may be regarded as the limiting case of two large equal and opposite point charges at a small distance apart. Volume and surface distribution of dipoles also play an important part in the theory. The study of Electrostatics is based primarily on columb‟s inverse square law. BASIC DEFINITIONS: DEFINITION : 1.1 A quantity which has both magnitude as well as direction is called vector.                                                                                 direction                                                       magnitude   DEFINITION: 1.2 ❖ A scalar is a physical quantity with magnitude only. ❖ A vector is a physical quantity with magnitude and direction. ❖ A unit vector has magnitude one. ❖ The position vector r = (x, y, z) DEFINITION : 1.3             It is branch of mechanics which deals with the motion of bodies under the action of given forces. DEFINITION : 1.4           The term ‘particle’ we mean that a finite mass occupying a point in the Eulidean space. In fact this is a purely mathematical concept.           In physics by the term ‘particle’ we mean a mass occupying an infinitesimally small amount of volume approximately compared to the distance between masses. Example :            The earth and the moon may be thought of as particles approximately, compared to their distance apart. DEFINITION : 1.5           When a constant force F acts upon a particle for a time – interval    then the product     is called the impulse.                     Then                  If the force acts from a time t1 to a time t2 then the change in momentum during this time – interval is                           DEFINITION : 1.6           Work is said to be done by a force when its point of application undergoes a displacement. DEFINITION : 1.7           In physics, motion is the change in the position of an object over time.           Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and time.    DEFINITION : 1.8           A measure of how much matter is in an object mass is measured in grams and pounds. DEFINITION: 1.9                  [work done(w) by the force   during the displacement    the scalar quantity] DEFINITION : 1.10            The rate change of displacement with respect to time is known as velocity. It is a vector quantity because is has both magnitude and direction velocity is denoted by  . The Inverse Square Law : 2.1  If two electric charges e1, e2 are placed in vacuo at a distance r apart, the force between them is proportional to          Further if the charges are of the same sign, the force is repulsive if they are of opposite signs, the force as attractive.  Thus the force F on a charge e2 at Q due to another charge e1 at P is given by                                                                                                                                                                                     e2      Q                                                                                                                      r                   P   e1  Where C is a constant and r is the position – vector of a relative to P. In C.G.S electrostatic unit system, we define a unit charge as one which, in vecuo, repels an exactly similar charge at a distance of one cm with a force of one dyne. In this system c = 1 and             ……………….(1)  This gives both magnitude and direction of the force, for if e1, e2 ar of like signs, the force on Q is refulsive, if of opposite signs, the force is attractive.                                                                                     P    (x, y, z)                                                            ri                                                                                                              Q                                                                       We define the electrostatic intensity or the electrostatic field vector at a point as the force that would act in vacuo on a unit charge placed at that point, due to the rest of the charges, provided the unit test charge is assumed not to disturb the field due to the other charges. This intensity has at each point a definite magnitude and a definite direction and is thus a vector. We denoted it by E. Thus for n charges           e1, e2, ………,en               ………………………(2)  Where ri denotes the position vector of P relative to the ith charge ei.  Now                                                                             If we define             ………………..(3)                      Then                                                 …………………….(4)                    Thus further from (2)                                         ……………………(5)                  E is a solenoidal vector.  Also                      …………………….(6)  E is also an irrotational vector.  Combining (4) and (5) we have          Or        ………………(7)  Therefore at a point not occupied by charges   satisfied Laplace’s equation (7)  We can verify (6) in an alternative manner. For any closed path, that does not include a charge, we have                                      ………………….(8)  But                 Represent the work done by the electric intensity, when a unit positive charge goes round a closed path. Thus electrostatic intensity is a conservative force and the work done in carrying a unit positive charge from one position to another does not depend on the path, but only on the position – vector of the end point.  Now                                   As from (3)          Thus   can now be interpreted as the work done by the electrostatic intensity in carrying unit positive charge from P to infinity along any path what so ever or as the work done against the electrostatic force in bringing the unit positive charge from infinity to P. Some Fundamental Definitions :  DEFINITION : 2.1  The scalar point function   defined in              is called the electrostatic potential of the field due to the given system of charges. The sufaces                    For various values of the constant constitute a family of equipotantial surfaces. The vector equation of the family is                    ………………..(1)  Which shows that no work is done in going from one point to another along an equipotential surface. DEFINITION : 2.2  A line of force is a curve such that the tangent to it at any point gives the direction of E at the point.  The vector differential equation of the lines of force is  dr x E = 0  ………………….(2)  Through every point in space in general, only one line of force will pass, but when E = 0 the direction of the line of force becomes indeterminate such a point is called a nalltral point or a point of equilibrium. DEFINITION : 2.3  The flux of a vertor E across the surface element da is defined as the scalar E. da If S is any surface, closed    open the flux of E across S is                                                                    s   If the surface is closed the flux of E out of S is                                                                      Where the direction of da is that of the outward drawn normal.  The flux of E out of the surface of a sphere of radius a, at the centre of which is placed a charges e, is                                                                                      Gauss’s Theorem :2.2  Statement :  If N is the flux of the electrostatic intensity E out of any closed surface S , then                    …………………….(1)                                          Where Q denotes the total charge enclosed by the surface S. Proof :                                                                                                                                                 Where    is the solid angle subtended by the closed surface at the ith charge. Now from,  (i) At an internal point. The whole area of the unit sphere = 4  (ii) At an point on the surface : surface area of half the unit sphere = 2   (iii) At an external point : solid angle subtended is zero.  And   =    (or) 0 according as e1 is inside or outside the surface.  Hence the theorem follows.  Note :1  For a point charge, if any, on the surface S itself,   = 2  and in this case, the theorem becomes N=4           ………………….(2)  Where    denotes the total charge on the surface. Note :2  We have proved Gauss’s theorem on the hypothesis that all the charges are discrete. But if the charge distribution is continuous the contribution to N by an element dv of charge is 4      or zero according as the charge is inside or outside S. Integration through all the space occupied by the charge distribution proves the theorem.    Theorem: 2.3  The strength of a tube of force (or) the flux across any section of it) remains constant along its path.                                                                                       d                                                                                    E2                      d        E2  As there is no flux out of a tube of force, applying Gauss’s theorem to the closed surface show in the figure.                         E1 . da + E2 . da2 = 0  E1 ds1 = E2 ds2 Uniqueness of the solution of an electrostatic problem : 2.4 Problem :2.4.1  Let    and    be the two possible potential distributions due to given charge distribution   and surface distribution   on the surface of given conductors. Then in space, we have                 and on the surface of conductors, we have                and that if          Then       in all space  and      on the surface of all conductors. Now from divergence theorem  Now from Divergence theorem                                                    …………………(1)  But       in all space and       on the surface of the conductors.  Also from considerations of magnitude the surface integral vanishes over the sphere at infinity. /          dv = 0                                                         v             Which implies that      in all space.             Or                 But       over the surfaces of conductors        in all space.  Thus the potential distribution must be unique. Coulomb’s Law : 2.5  Consider a charge q at rest with position vector      . This is called the ‘source’ charge. When a test charge Q with position vector    is placed at a distance    away, the force is given by coulomb’s Law. or                                                                                                      Q                                                                                                                                                       r                                      q       Fig (a)                                                Coulomb showed experimentally that a force on the point charge Q due to the single point charge q at rest is 1) Proportional to the product of (Their magnitudes) of the charge. 2) Inversely proportional to the square of the distance separating them ( ). Fig (a) 3) Acts along the line joining the two charges                  ……………..(1)     is called the permittivity of free space                       = unit vector in the direction of     is the separation vector from          (test charge Q)  i.e.       r is its magnitude.                The force is directed from q to Q. If the force is repulsive the charges have same sign and is of oppositive sign if the force is attractive. Principle of Superposition : 2.6             The principle states that the interaction between any two charge is completely unaffected by the presence of other charges. To determine the force on Q, we find the force F1 due to q1 alone (ignoring all the other charges) then we find the force F2 due to q2 alone and so on.             The total force    is the vector sum of the individual forces.                                                                                                                                                                                                                                                                                                                       are distances of charges               from and              are unit vectors in the direction of              respectively.         The Electric Field : 2.7            If we place a charge Q at a distance   from a charge q, it experiences a force. Hence the region surrounding the charge q in which it exerts an electric force on another charge Q is termed as the electric field of charge q.             The intensity of the electric field at a given point is defined as the electric force experienced by a unit positive charge placed at that point.            If a positive charge q experiences a force   , then the electric field intensity is                    Or in general              ……………………(1)             The electric field intensity    at a point is defined as the limiting value of the ratio of the force    experienced by a charge q placed at that point, as the value of q tends to zero.             ………………….(2)             For several point charges                                          …………………….(3) Concept of a field :2.8                   Fields are not composed of matter (electrons, protons and neutrons etc), but they be still exist. Fields can be considered as a complementary component of the world to the particles of matter. Particles are localised with a relatively well – defined position whereas fields extend throughout all of space.           ‘Fields are found to affect matter, and matter to affect fields, and it is through their mutual effects that we determine the properties of both matter and fields. For example, consider two particles X and Y. we can imagine the interaction between x and y as a three stage process :          Particle x affects the field around it.  The disturbance of the field around x spreads away from x.             A little later, particle y is affected by the disturbance of the field created by x.                 Notice that the field itself is not an observable ‘object’ we can infer its properties from systematic studies of the way one particle affects other particles around it. Currently we can explain all observable phenomena in the universe in term of three different types of fields. The gravitational, electro weak and strong fields. It is hoped that all these fields   will be understood on the basis of a ‘unified’ field. In coulomb’s law, the force exerted by one particle on another is seen as an action at a distance without the in between space. But there is a finite time required for electromagnetic interactions to be propagated. This time is the time for the speed of light to travel in the given medium. Thus the force is not transmitted in stantaneously with infinite speed but requires some time.       CONCLUSION :          The study of this project is vector Analysis in applications of vector calculus. I have considered, work, D’Alemberts principle and General Equation of motion of a Rigid body of vector calculus. Wheel related calculus shell related algorithm, theorem, problem and some operations of applications of vector calculus. Though this work deal with only basic ideas and concept may be extended in the application of some field.    References:     1. Absos Ali Shaisk Sanjib Kumar Jana “ Vector Analysis with applications, Naroso publishing house New Delhi. 2. Shanti Narayan and P.K.Mittal : A Text Book of vector calculus (with applications), First Edition – 1958. 3. Ram Krishna Ghosh Kantish Chandra Maity : Vector Analysis, First Edition – 1958. 4. M.D.Rasinghana : Dynamics, First Edition – 2006. 5. B.V.Ramana : Applied Mathematic –I, First Edition – 2007.                                                                                 

A STUDY ON ELECTRO STATICS

A  STUDY ON   ELECTRO STATICS (P.ELAVARASI,RRAMYA,M.MAHALAKSHMI,Dr.S.SANGEETHA) (ramya25071987@gmail.com,sangeethasankar2016@gmailcom) Department of Mathematics Dhanalakshmi Srinivasan College of  Arts and Science for Women (Autonomous) Perambalur ABSTRACT:                                   In  this  paper we are elaborate Electrostatics is based primarily inverse law. Then small volume distance between two forces.  Also  we  solve the Uniqueness of the solution of an electrostatic problem .  Keywords:   Electro , vector , volume , surface  , direction ,magnitude Introduction: Vector calculus, or vector analysis is a branch of mathematics concerned with differentiation and integration of vector fields. “Vector calculus” is some times used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Electrostatics is concerned with electricity at rest. This electricity may be in the form of discrete point charges or in the form of a continuous charge distribution with volume density 𝜌 or surface density 𝜎 or even in the form of electric dipoles, where a dipole may be regarded as the limiting case of two large equal and opposite point charges at a small distance apart. Volume and surface distribution of dipoles also play an important part in the theory. The study of Electrostatics is based primarily on columb‟s inverse square law. BASIC DEFINITIONS: DEFINITION : 1.1 A quantity which has both magnitude as well as direction is called vector.                                                                                 direction                                                       magnitude   DEFINITION: 1.2 ❖ A scalar is a physical quantity with magnitude only. ❖ A vector is a physical quantity with magnitude and direction. ❖ A unit vector has magnitude one. ❖ The position vector r = (x, y, z) DEFINITION : 1.3             It is branch of mechanics which deals with the motion of bodies under the action of given forces. DEFINITION : 1.4           The term „particle‟ we mean that a finite mass occupying a point in the Eulidean space. In fact this is a purely mathematical concept.           In physics by the term „particle‟ we mean a mass occupying an infinitesimally small amount of volume approximately compared to the distance between masses. Example :            The earth and the moon may be thought of as particles approximately, compared to their distance apart. DEFINITION : 1.5           When a constant force F acts upon a particle for a time – interval ∆𝑡 then the product 𝐹∆𝑡 is called the impulse.                     Then 𝐹 =𝑑𝑝𝑑𝑡           If the force acts from a time t1 to a time t2 then the change in momentum during this time – interval is 𝑃2 − 𝑃1 =   𝑃2𝑃1𝑑𝑝 =   𝑡1𝑡2𝐹. 𝑑𝑡 DEFINITION : 1.6           Work is said to be done by a force when its point of application undergoes a displacement. DEFINITION : 1.7           In physics, motion is the change in the position of an object over time.           Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and time.    DEFINITION : 1.8           A measure of how much matter is in an object mass is measured in grams and pounds. DEFINITION: 1.9 𝑊 = 𝐹. 𝐴𝐵           [work done(w) by the force 𝐹 during the displacement 𝐴𝐵 the scalar quantity] DEFINITION : 1.10            The rate change of displacement with respect to time is known as velocity. It is a vector quantity because is has both magnitude and direction velocity is denoted by 𝑣. The Inverse Square Law : 2.1  If two electric charges e1, e2 are placed in vacuo at a distance r apart, the force between them is proportional to  𝑒1𝑒2𝑟2  Further if the charges are of the same sign, the force is repulsive if they are of opposite signs, the force as attractive.  Thus the force F on a charge e2 at Q due to another charge e1 at P is given by 𝐹 = 𝐶 𝑒1𝑒2𝑟2 𝑟𝑟= 𝑐𝑒1𝑒2𝑟3𝑟                                                                                                                                                              e2      Q                                                                                                                      r                   P   e1  Where C is a constant and r is the position – vector of a relative to P. In C.G.S electrostatic unit system, we define a unit charge as one which, in vecuo, repels an exactly similar charge at a distance of one cm with a force of one dyne. In this system c = 1 and  𝐹 =  𝑒1𝑒2𝑟3𝑟 ……………….(1)  This gives both magnitude and direction of the force, for if e1, e2 ar of like signs, the force on Q is refulsive, if of opposite signs, the force is attractive.                                                                                     P    (x, y, z)                                                            ri                                                                                                              Q                                                                       We define the electrostatic intensity or the electrostatic field vector at a point as the force that would act in vacuo on a unit charge placed at that point, due to the rest of the charges, provided the unit test charge is assumed not to disturb the field due to the other charges. This intensity has at each point a definite magnitude and a definite direction and is thus a vector. We denoted it by E. Thus for n charges           e1, e2, ………,en 𝐸 =  𝑒𝑖𝑟𝑖3 𝑟𝑖𝑛𝑖=1 ………………………(2)  Where ri denotes the position vector of P relative to the ith charge ei.  Now 𝑟𝑖2 =  𝑥 − 𝜀𝑖 2 +  𝑦 − 𝑣𝑖 2 +  𝑧 − 𝜁𝑖 2 ∇ 1𝑟𝑖 = −1𝑟𝑖2  𝑥 − 𝜀𝑖𝑟𝑖𝑖 +𝑦 − 𝜂𝑖𝑟𝑖𝑗 +𝑧 − 𝜁𝑖𝑟𝑖𝑘  = −1𝑟𝑖3 𝑟𝑖   If we define ∅ =  𝑒𝑖𝑟𝑖𝑛𝑖=1  ………………..(3)                      Then ∇∅ =  𝛻  𝑒𝑖𝑟𝑖 𝑛𝑖=1= − 𝑒𝑖𝑟𝑖3 𝑟𝑖𝑛𝑖=1= −𝐸 𝐸 = −𝛻∅ = −𝑔𝑟𝑎𝑑∅   …………………….(4)                    Thus further from (2) 𝑑𝑖𝑣𝐸 = 𝛻  𝑒𝑖𝑟𝑖3 𝑟𝑖𝑛𝑖=1  =  𝑒𝑖𝛻  𝑟𝑖𝑟𝑖3 𝑛𝑖=1 = 0  ……………………(5)                  E is a solenoidal vector.  Also 𝑐𝑢𝑟𝑙 𝐸 = −∆ 𝑥  ∆∅ = 0   …………………….(6)  E is also an irrotational vector.  Combining (4) and (5) we have ∇ 𝛻∅ = 0  Or ∇2∅ = 0  ………………(7)  Therefore at a point not occupied by charges ∅ satisfied Laplace‟s equation (7)  We can verify (6) in an alternative manner. For any closed path, that does not include a charge, we have  𝐸. 𝑑𝑟 = − 𝛻∅ 𝑑𝑟  =  −    𝑑∅ = 0 𝑐𝑢𝑟𝑙 𝐸 = 0  ………………….(8)  But           𝐸. 𝑑𝑟  Represent the work done by the electric intensity, when a unit positive charge goes round a closed path. Thus electrostatic intensity is a conservative force and the work done in carrying a unit positive charge from one position to another does not depend on the path, but only on the position – vector of the end point.  Now    ∞𝑃𝐸. 𝑑𝑟 = −  ∞𝑃𝑑∅ = ∅ 𝑃 − ∅ ∞ = ∅(𝑃)  As from (3)  ∅ ∞ = 0  Thus ∅ can now be interpreted as the work done by the electrostatic intensity in carrying unit positive charge from P to infinity along any path what so ever or as the work done against the electrostatic force in bringing the unit positive charge from infinity to P. Some Fundamental Definitions :  DEFINITION : 2.1  The scalar point function ∅ defined in ∅ =   𝑛𝑖=1𝑒𝑖𝑟𝑖 is called the electrostatic potential of the field due to the given system of charges. The sufaces ∅ 𝑥, 𝑦, 𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡  For various values of the constant constitute a family of equipotantial surfaces. The vector equation of the family is 𝑑∅ = 𝑑𝑟. 𝛻∅ = 0 𝑑𝑟. 𝐸 = 0  ………………..(1)  Which shows that no work is done in going from one point to another along an equipotential surface. DEFINITION : 2.2  A line of force is a curve such that the tangent to it at any point gives the direction of E at the point.  The vector differential equation of the lines of force is  dr x E = 0  ………………….(2)  Through every point in space in general, only one line of force will pass, but when E = 0 the direction of the line of force becomes indeterminate such a point is called a nalltral point or a point of equilibrium. DEFINITION : 2.3  The flux of a vertor E across the surface element da is defined as the scalar E. da If S is any surface, closed 𝜍𝑟 open the flux of E across S is     𝐸. 𝑑𝑎                                                           s   If the surface is closed the flux of E out of S is    𝑠𝐸. 𝑑𝑎                                                             Where the direction of da is that of the outward drawn normal.  The flux of E out of the surface of a sphere of radius a, at the centre of which is placed a charges e, is    𝑠𝐸. 𝑑𝑎 =    s𝑒𝑎24𝜋𝑎2 = 4𝜋𝑒                                                             Gauss’s Theorem :2.2  Statement :  If N is the flux of the electrostatic intensity E out of any closed surface S , then 𝑁 =    𝑠  𝐸. 𝑑𝑎 = 4𝜋𝑄   …………………….(1)                                          Where Q denotes the total charge enclosed by the surface S. Proof :    s𝐸. 𝑑𝑎 =    s  𝑛𝑖=1𝑒𝑖𝑟𝑖𝑟𝑖3  𝑑𝑎                                                                =   𝑛𝑖=1𝑒𝑖     𝑟𝑖𝑟𝑖3  𝑑𝑎 =   𝑛𝑖=1𝑒𝑖     𝑑𝜔𝑖 =   𝑛𝑖=1𝑒𝑖𝜔𝑖   Where 𝜔𝑖  is the solid angle subtended by the closed surface at the ith charge. Now from,  (i) At an internal point. The whole area of the unit sphere = 4𝜋 (ii) At an point on the surface : surface area of half the unit sphere = 2 𝜋 (iii) At an external point : solid angle subtended is zero.  And 𝜔𝑖= 4𝜋 (or) 0 according as e1 is inside or outside the surface.  Hence the theorem follows.  Note :1  For a point charge, if any, on the surface S itself, 𝑤𝑖= 2𝜋 and in this case, the theorem becomes N=4𝜋𝑄 + 2 𝜋𝑄′   ………………….(2)  Where 𝑄′ denotes the total charge on the surface. Note :2  We have proved Gauss‟s theorem on the hypothesis that all the charges are discrete. But if the charge distribution is continuous the contribution to N by an element dv of charge is 4 𝜋𝜌𝑑𝑣 or zero according as the charge is inside or outside S. Integration through all the space occupied by the charge distribution proves the theorem.    Theorem: 2.3  The strength of a tube of force (or) the flux across any section of it) remains constant along its path.                                                                                       dƩ2                                                                                  E2                      dƩ1      E2  As there is no flux out of a tube of force, applying Gauss‟s theorem to the closed surface show in the figure.                         E1 . da + E2 . da2 = 0  E1 ds1 = E2 ds2 Uniqueness of the solution of an electrostatic problem : 2.4 Problem :2.4.1  Let ∅1 and ∅2 be the two possible potential distributions due to given charge distribution 𝜌 and surface distribution 𝜍 on the surface of given conductors. Then in space, we have ∇2∅1 = 𝛻2∅2 = −4𝜋𝜌  and on the surface of conductors, we have ∇∅1 = 𝛻∅2 = −4𝜋𝜍𝑛   and that if 𝜃 = ∅1 − ∅2  Then ∇2𝜃 = 0 in all space  and ∇𝜃 = 0 on the surface of all conductors. Now from divergence theorem  Now from Divergence theorem      𝜃∇𝜃   . 𝑑𝑎 =      𝑑𝑖𝑣 𝜃𝛻𝜃 . 𝑑𝑣 =     𝜃𝛻2𝜃 + (𝛻𝜃)2 𝑑𝑣 …………………(1)  But ∇2𝜃 = 0 in all space and ∇𝜃 = 0  on the surface of the conductors.  Also from considerations of magnitude the surface integral vanishes over the sphere at infinity. /     (𝛻𝜃)2dv = 0                                                         v             Which implies that ∇𝜃 = 𝜃 in all space. 𝜃 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. Or ∅1 − ∅2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡  But ∅1 = ∅2 over the surfaces of conductors  ∅1 = ∅2 in all space.  Thus the potential distribution must be unique. Coulomb’s Law : 2.5  Consider a charge q at rest with position vector 𝑟1    . This is called the „source‟ charge. When a test charge Q with position vector  𝑟  is placed at a distance 𝑟   away, the force is given by coulomb‟s Law. or                                                                                                      Q                                                                                         𝑟                                                             r                                      q       Fig (a)                                                Coulomb showed experimentally that a force on the point charge Q due to the single point charge q at rest is 1) Proportional to the product of (Their magnitudes) of the charge. 2) Inversely proportional to the square of the distance separating them (𝑟 ). Fig (a) 3) Acts along the line joining the two charges  𝐹 =14𝜋𝜀0𝑞𝑄𝑟2. 𝑟  ……………..(1) ∈ 0 is called the permittivity of free space = 8.85 𝑥 10−12𝑐2𝑁. 𝑚2 𝑟 = unit vector in the direction of 𝑟  𝑟  is the separation vector from 𝑟  𝑞 𝑡𝑜 𝑟  (test charge Q)  i.e. 𝑟 = 𝑟 − 𝑟  r is its magnitude.                The force is directed from q to Q. If the force is repulsive the charges have same sign and is of oppositive sign if the force is attractive. Principle of Superposition : 2.6             The principle states that the interaction between any two charge is completely unaffected by the presence of other charges. To determine the force on Q, we find the force F1 due to q1 alone (ignoring all the other charges) then we find the force F2 due to q2 alone and so on.             The total force 𝐹  is the vector sum of the individual forces. 𝐹 = 𝐹1    + 𝐹2     + 𝐹3     + ⋯… . 𝐹𝑛      𝐹 =14𝜋𝜀0 𝑞1𝑄𝑟12 𝑟1 +𝑞2𝑄𝑟22 𝑟2 + ⋯…… .𝑞𝑛𝑄𝑟𝑛2 𝑟𝑛                                                                                                                                                                                                                      𝑟1    , 𝑟2    …… . . 𝑟𝑛     are distances of charges 𝑞1 , 𝑞2 …………… . 𝑞𝑛  from and 𝑟1 , 𝑟2 ………… . 𝑟𝑛  are unit vectors in the direction of 𝑟1 , 𝑟2 ………… . 𝑟𝑛  respectively.         The Electric Field : 2.7            If we place a charge Q at a distance 𝑟  from a charge q, it experiences a force. Hence the region surrounding the charge q in which it exerts an electric force on another charge Q is termed as the electric field of charge q.             The intensity of the electric field at a given point is defined as the electric force experienced by a unit positive charge placed at that point.            If a positive charge q experiences a force 𝐹 , then the electric field intensity is 𝐸 =𝐹 𝑞             Or in general 𝐸 =𝐹 𝑞       ……………………(1)             The electric field intensity 𝐸  at a point is defined as the limiting value of the ratio of the force 𝐹  experienced by a charge q placed at that point, as the value of q tends to zero. 𝐸 =𝑞4𝜋𝜀0𝑟2𝑟 ………………….(2)             For several point charges 𝑞1, 𝑞2 …………… . 𝑞𝑛   𝐸  𝑟  =14𝜋𝜀0  𝑛𝑖=1𝑞𝑖𝑟𝑖2 𝑟𝑖   …………………….(3) Concept of a field :2.8                   Fields are not composed of matter (electrons, protons and neutrons etc), but they be still exist. Fields can be considered as a complementary component of the world to the particles of matter. Particles are localised with a relatively well – defined position whereas fields extend throughout all of space.           „Fields are found to affect matter, and matter to affect fields, and it is through their mutual effects that we determine the properties of both matter and fields. For example, consider two particles X and Y. we can imagine the interaction between x and y as a three stage process :          Particle x affects the field around it.  The disturbance of the field around x spreads away from x.             A little later, particle y is affected by the disturbance of the field created by x.                 Notice that the field itself is not an observable „object‟ we can infer its properties from systematic studies of the way one particle affects other particles around it. Currently we can explain all observable phenomena in the universe in term of three different types of fields. The gravitational, electro weak and strong fields. It is hoped that all these fields ÷ will be understood on the basis of a „unified‟ field. In coulomb‟s law, the force exerted by one particle on another is seen as an action at a distance without the in between space. But there is a finite time required for electromagnetic interactions to be propagated. This time is the time for the speed of light to travel in the given medium. Thus the force is not transmitted in stantaneously with infinite speed but requires some time.       CONCLUSION :          The study of this project is vector Analysis in applications of vector calculus. I have considered, work, D‟Alemberts principle and General Equation of motion of a Rigid body of vector calculus. Wheel related calculus shell related algorithm, theorem, problem and some operations of applications of vector calculus. Though this work deal with only basic ideas and concept may be extended in the application of some field.    References:     1. Absos Ali Shaisk Sanjib Kumar Jana “ Vector Analysis with applications, Naroso publishing house New Delhi. 2. Shanti Narayan and P.K.Mittal : A Text Book of vector calculus (with applications), First Edition – 1958. 3. Ram Krishna Ghosh Kantish Chandra Maity : Vector Analysis, First Edition – 1958. 4. M.D.Rasinghana : Dynamics, First Edition – 2006. 5. B.V.Ramana : Applied Mathematic –I, First Edition – 2007.                                                                                 

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A STUDY ON ELECTRO STATICS

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