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Electrostatics Introduction:Electrostatics is a branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges. When we comb our hair on a dry day and bring the comb close to tiny pieces of paper, we note that they are attracted by the comb. This happens due to the transfer of the electron from hair to the comb. Now, the comb becomes negatively charged. So, it induces a dipole in the pieces of paper and the paper gets attracted. 2. Electric charge:Electric charge like mass is one of the fundamental attributes of the particles of which the matter is composed of. The charge is the physical property of certain fundamental particles (like electron and proton) by virtue of which they interact with the other similar fundamental particles. Two types: - There are two types of electric charge named by the American scientist Benjamin Franklin as positive charge and negative charge. Like charges repel each other while unlike charges attract. In nature, atoms are normally found with an equal number of protons (positively charged particles) and electrons (negatively charged particles). Thus atom is electrically neutral. The charge on an electron or a proton is the smallest amount of free charge that has been discovered. 1.6 10 19 C 19 Charge on electron: 1.6 10 C Charge on proton: Charges of larger magnitude are built up on an object by adding or removing electrons. If in a body there is excess of electrons over its natural configuration conventionally the body is said to be negatively charged and if there is deficiency of electrons it is said to be positively charged. Negative charged body: Body has gained electrons Positive charged body: Body has lost source electrons 2.1 Basic properties of electric charge Charge is a scalar quantity and can be of two types (i.e. +ve charge and –ve charge). It is added algebraically. Charge is invariant: It is independent of frame of reference. Charge is transferable: if a charged body is put in contact with an uncharged body, the uncharged body becomes charged due to transfer of electrons from one body to the other. Charge is always associated with mass: charge cannot exist without mass, though mass can exist without mass. So, the presence of charge itself is a convincing proof of existence of mass. Charge is quantized: Robert Millikan discovered that electric charge always occurs as some integral multiple of fundamental unit of charge (e). q=ne Where n is an integer and it corresponds to the number of electrons. Charge on a body can never be (1/3) e, (2/3) e etc. as it is due to transfer of electron. Charge is conserved. During any chemical, nuclear, decay etc., the net electric charge of an isolated system remains constant. In other words, charge can neither be created nor be destroyed. When charge particle is at rest it produces only electric field. When charged particle is accelerated it produces electric field ++ magnetic field ++ radiate energy. Insulators and conductors On the basis of electrostatic theory all substances can be divided into three main groups, 1. 2. 3. Conductors Insulators Semiconductors Conductors: An electrical conductor is a substance in which electrical charge carriers, usually electrons, move easily from atom to atom with the application of voltage. Example: Metals, earth, human body etc. Insulators: An insulator is a substance in which electrical charge carriers such as electrons, are tightly bound to their respective atoms. Example: Plastics, hard rubber, glass etc. Semiconductor: They are third class of materials and their electrical properties are somewhere between those of insulators and conductors. Example: Silicon (Si) and Germanium (Ge). 2.2 Charging of a body Charging of a body can be achieved by three methods, 1. 2. 3. Friction conduction induction 2.2.1 Charging by friction All material bodies contain a large number of electrons and an equal number of protons in their normal state. When rubbed against each other, electrons gains energy and some electrons are transferred from one body to another. The body that donates the electron becomes positively charged while that which receives the electrons becomes negatively charged. For example: When a glass rod is rubbed with a silk cloth, glass rod becomes positively charged because it donates the electrons while the silk cloth becomes negatively charged because it receives electrons. 2.2.2 Charging by contact The process of giving one object a net electric charge by placing it in contact with another object that is already charged is known as charging by contact. When a negatively charged ebonite rod is touched with a metal object, such as a sphere, some of the excess electrons from the rod are transferred to the sphere. Once the electrons are on the metal sphere, where then can move readily, they repel each other and spread out over the sphere's surface. The insulated stand prevents them from flowing to the earth. When the rod is removed, the sphere is left with a negative charge distributed over its surface. 2.2.3 Charging by induction Induction is defined as the redistribution of electrical charge in an object caused by the influence of nearby charges. Consider a negatively charged rod is brought close to a metal sphere without touching it. In the sphere, the free electron close to the rod moves to the other side by repulsion. As a result, the part of the sphere nearer to the rod becomes positively charged and the part farthest from the rod becomes negatively charged. However, the net on the rod is still conserved i.e. zero. Now if the rod is removed, the free electrons return to their original position and the charged regions disappear. Now, when a metal wire is attached to the sphere and the ground as shown in figure (b), some of the free electrons leave the sphere and flows into the earth. Now if the earthing wire is removed, followed by the charged rod, then the sphere is left with a net positive charge as shown in figure (c). 3.0 Coulomb's law Coulomb's law was discovered by Charles Augustin de Coulomb in the year 1785. Coulomb's law describes how static charges interact with one another. Coulomb's law stated that the electrostatic force between two stationary charges is proportional to the product of magnitude of charges and inversely proportional to the square of the distance between them. Mathematically, F q1 q 2 d2 q1q2 40 d 2 qq Or F 1 2 2 d Where q1 charge on particle 1, q2 Charge on particle 2 12 2 1 2 d = distance between particles, 0 8.85 10 C N m =Permittivity of free space 1 Or F and 9 10 9 N 1 m 2 C 2 The electrostatic force is an action-reaction pair, i.e., the two charges exert equal and opposite forces on each other. FAB FBA 3.1 Coulomb's law in vector form Coulomb's law can be written as, F q1q 2 r 40 r 3 1 For the above configuration of charged particles, the Coulomb's law can be written as F q1q 2 r2 r1 40 r2 r1 3 1 4.0 Principle of superposition: The interaction between any two charges is independent of the presence of all other charges. The electrostatic force is a vector quantity. Therefore, the net force on any one charge is the vector sum of all the forces exerted on it due to each of the other charges interacting with it independently. The principle of superposition of forces holds good for any number of charges. So total charge on q is – FNet F1 F2 F3 ............... Fn 5.0 Continuous charge distribution Depending on the dimension of a body, three types of continuous charge distribution are, 1-D body (rod): Linear charge density 2-D body (film or lamina): Surface charge density 3-D body: Volume charge density 5.1 Linear charge density Charge per unit length is known as linear charge density. It is denoted by symbol λ. ch arg e length 1 Its SI unit is Cm . It is used for 1-D body. Example: Rod 5.2 Surface charge density Charge per unit area is known as surface charge density. It is denoted by symbol σ Its SI unit is Cm 2 ch arg e Area . It is used for 2-D body. Example: Lamina, film, thin plate, spherical shell, cylindrical shell etc. 5.3 Volume charge density Charge per unit volume is known as volume charge density. It is denoted by symbol ρ. Its SI unit is ch arg e Volume Cm 3 It is used for 3-D body. Example: Solid cone, solid sphere, solid cylinder etc. 6.0 Electric field Electric field due to a point charge is the space surrounding it, within which an electric force can be experienced by any other charge. It is represented by E. Consider a positive charge Q is placed at any point in an electric field and it experiences a force F, then the electric field is defined as, E F Q The magnitude of E is the force per unit charge and its direction is that of FF (i.e., when the force acts on a positive charge) The direction of E is opposite to that of F when the force acts on a negative charge. The electric field is a vector quantity. Mathematically it is expressed as, F E Q SI unit of an electric field is NC 1 . Its dimensional formula is MLT 3 A 1 6.1 Electric field due to a point charge The electrical force F between a positive point charge q and test charge q0 at point P which is at a distance r is given by, F qq0 r2 Then the electric field due to positive point charge q at a distance r is, E Or E F q0 q r2 q E 2r r If q is positive, then E is directed away from q. Or in vector form If q is negative, then E is directed towards q. Electric Dipole:An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance of 2a. o Direction from –q to q is the direction of the dipole. o The mid-point of locations of –q and q is called the center of the dipole. o Total charge of an electric dipole is zero but since the charges are separated by some distance the electric field do not cancel out. o Dipole moment is the mathematical product of the separation of the ends of a dipole and the magnitude of the charges (2a x q). o Some molecules like H2O, have permanent dipole moment as their charges do not coincide. These molecules are called polar molecules. o Permanent dipoles have a dipole moment irrespective of any external Electric field. 7.0 Electric field lines An electric field line is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction of the electric field vector at that point. In other words, it is a way of pictorially mapping the electric field around a configuration of charges. The relative closeness of the lines at some place give an idea about the intensity of electric field at that point. 7.1 Properties of electric field lines E at that point. This is also the path to which a 1. The tangent to a line at any point gives the direction of positive test charge will tend to move if free to do so. 2. Electric field lines always begin with a positive charge and end on a negative charge and do not start or stop in mid space. 3. The number of lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge. 4. Two lines can never intersect. If it happens then two tangents can be drawn at the point of intersection. It means two directions of the electric field at that point which is not possible. 5. In a uniform field, the field lines are straight, parallel and uniformly spaced. 6. The electric field lines can never form closed loops as a line can never start and end on the same charge. 7. Electric field lines also give us an indication of the equipotential surface (a surface which has the same potential). 8. Electric field lines always flow from higher potential to lower potential. 9. In a region where there is no electric field, lines are absent. This is why inside a conductor (where an electric field is zero) there cannot be any electric field line. 10. Electric field lines start or end normally from the surface of a conductor. 7.2 Electric flux Electric flux is a measure of the number of electric field lines crossing a surface. If the electric field lines pass through a surface then the surface is said to have flux linked with it. It is given by, d E.ds Where, E =electric field and ds =Area vector of small area element. The area vector of a closed surface is always in the direction of outward drawn normal. So, the total flux lined with whole of the body is the closed integral of equation E.ds Where, ∮: Represents closed integral done for a closed surface. 2 C 1 . The SI unit of electric flux is Nm Its dimensional formula is Electric flux is a scalar quantity as it is a dot product of two vector quantity. Electric flux is zero for a surface when electric field field is parallel to the surface. ML T 3 3 A1 E is perpendicular to the area vector ds or electric 8.0 Gauss's law Gauss's law states that the total electric flux through an imaginary closed surface is proportional to the total electric charge enclosed within the surface. In other words, the net electric flux through any closed surface is equal to the net charge inside the surface divided by 0 . Mathematically it can be written as, e qin 0 Or qin E.ds 0 If the area of the enclosed surface is known then the above equation is reduced as, ES qin 0 Where, qin Net charge inside the surface E Electric field at any point on the surface s Electric flux through any closed surface S: Area of the enclosed surface where electric field is perpendicular to the surface. Note: This technique is useful for calculating the electric field in situations where a degree of symmetry is high. Some important aspects of application of Gauss's theorem for determination of electric field 1. The imaginary surface which is chosen for application of Gauss's law is called a Gaussian surface. Any Gaussian surface can be chosen but it should not pass through a discrete charge. However, it can pass through a continuous charge distribution. 2. Gauss's theorem is mostly used for symmetric charge distribution. 3. The term 4. Gauss's theorem is based on the inverse square dependence on distance contained in the Coulomb's law. Any violation of Gauss's theorem will indicate the departure from the inverse square law. qin corresponds to sum of all charges enclosed within the Gaussian surface. 9.1 Electric field due to a point charge Let a point charge q is place at point O. So, for calculating the electric field at the point PP which is at a distance r from point O, we will draw a symmetrical Gaussian spherical surface of radius r from the charge. So, at every point, the electric field has the same magnitude EE and it is perpendicular to the surface. So, we can apply Gauss's law as, ES qin 0 E (4r 2 ) E qin 0 qin 40 r 2 qin r2 We get same equation for electric field due to a point charge q at a distance r from Coulomb's law.