Electrostatics: Properties of Charges, Coulomb's Law, and Electric Fields, Slides of Sociology

Download Electrostatics: Properties of Charges, Coulomb's Law, and Electric Fields and more Slides Sociology in PDF only on Docsity! ELECTROSTATICS - I – Electrostatic Force 1. Frictional Electricity 2. Properties of Electric Charges 3. Coulomb’s Law 4. Coulomb’s Law in Vector Form 5. Units of Charge 6. Relative Permittivity or Dielectric Constant 7. Continuous Charge Distribution i) Linear Charge Density ii) Surface Charge Density iii) Volume Charge Density Frictional Electricity: Frictional electricity is the electricity produced by rubbing two suitable bodies and transfer of electrons from one body to other. . + + + + + + + + + + + + - - - - - - - - - - + + + + + + + + + + Glass Silk Ebonite Flannel Electrons in glass are loosely bound in it than the electrons in silk. So, when glass and silk are rubbed together, the comparatively loosely bound electrons from glass get transferred to silk. As a result, glass becomes positively charged and silk becomes negatively charged. Electrons in fur are loosely bound in it than the electrons in ebonite. So, when ebonite and fur are rubbed together, the comparatively loosely bound electrons from fur get transferred to ebonite. As a result, ebonite becomes negatively charged and fur becomes positively charged. Note: Recently, the existence of quarks of charge ⅓ e and ⅔ e has been postulated. If the quarks are detected in any experiment with concrete practical evidence, then the minimum value of ‘quantum of charge’ will be either ⅓ e or ⅔ e. However, the law of quantization will hold good. Coulomb’s Law – Force between two point electric charges: The electrostatic force of interaction (attraction or repulsion) between two point electric charges is directly proportional to the product of the charges, inversely proportional to the square of the distance between them and acts along the line joining the two charges. Strictly speaking, Coulomb’s law applies to stationary point charges. r q1 q2F α q1 q2 F α 1 / r2 or F α q1 q2 r2 F = k q1 q2 r2 or where k is a positive constant of proportionality called electrostatic force constant or Coulomb constant. In vacuum, k = 1 4πε0 where ε0 is the permittivity of free space In medium, k = 1 4πε where ε is the absolute electric permittivity of the dielectric medium The dielectric constant or relative permittivity or specific inductive capacity or dielectric coefficient is given by F = q1 q2 r2 1 4πε0 In vacuum, F = q1 q2 r2 1 4πε0εr In medium, ε0 = 8.8542 x 10-12 C2 N-1 m-2 = 8.9875 x 109 N m2 C-2 1 4πε0 or = 9 x 109 N m2 C-2 1 4πε0 K = εr = ε ε0 Coulomb’s Law in Vector Form: r + q1 + q2 F21F12 r12 q1q2 > 0 q1q2 < 0 r + q1 - q2 F21F12 r12 In vacuum, for q1 q2 > 0, q1 q2 r2 1 4πε0 r21F12 = q1 q2 r2 1 4πε0 r12F21 = In vacuum, for q1 q2 < 0, q1 q2 r2 1 4πε0 r12F12 = q1 q2 r2 1 4πε0 r21F21 =& F12 = - F21 (in all the cases) r - q1 - q2 F21F12 r12 q1q2 > 0 Continuous Charge Distribution: Any charge which covers a space with dimensions much less than its distance away from an observation point can be considered a point charge. A system of closely spaced charges is said to form a continuous charge distribution. It is useful to consider the density of a charge distribution as we do for density of solid, liquid, gas, etc. (i) Line or Linear Charge Density ( λ ): If the charge is distributed over a straight line or over the circumference of a circle or over the edge of a cuboid, etc, then the distribution is called ‘linear charge distribution’. Linear charge density is the charge per unit length. Its SI unit is C / m. q λ = l λ = dq dl or + + + + + + + + + + + + dq dl Total charge on line l, q = ∫ λ dl l (ii) Surface Charge Density ( σ ): σ = q S σ = dq dS or If the charge is distributed over a surface area, then the distribution is called ‘surface charge distribution’. Surface charge density is the charge per unit area. Its SI unit is C / m2. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + dq dSTotal charge on surface S, q = ∫ σ dS S (iii) Volume Charge Density ( ρ ): ρ = q ז ρ = dq dז or If the charge is distributed over a volume, then the distribution is called ‘volume charge distribution’. Volume charge density is the charge per unit volume. Its SI unit is C / m3. Total charge on volume ז, q = ∫ ρ dז ז dq dז ELECTROSTATICS - II : Electric Field 1. Electric Field 2. Electric Field Intensity or Electric Field Strength 3. Electric Field Intensity due to a Point Charge 4. Superposition Principle 5. Electric Lines of Force i) Due to a Point Charge ii) Due to a Dipole iii) Due to a Equal and Like Charges iv) Due to a Uniform Field 6. Properties of Electric Lines of Force 7. Electric Dipole 8. Electric Field Intensity due to an Electric Dipole 9. Torque on an Electric Dipole 10.Work Done on an Electric Dipole Electric Field due to a Point Charge: O Z Y X + q0 + q F r Force exerted on q0 by q is q q0 r2 1 4πε0 rF = q q0 r3 1 4πε0 rF = P (x,y,z) Electric field strength is q0 F E = q r3 1 4πε0 E (r) = r or or r2 1 4πε0 E (r) = q r The electric field due to a point charge has spherical symmetry. If q > 0, then the field is radially outwards. If q < 0, then the field is radially inwards. E r20 Electric field in terms of co-ordinates is given by ( x2 + y2 + z2 ) 3/2 1 4πε0 E (r) = q i j k( x + y + z ) Superposition Principle: + q2 - q3 - q5 + q4 + q1 F12 F14 F15 F13 The electrostatic force experienced by a charge due to other charges is the vector sum of electrostatic forces due to these other charges as if they are existing individually. F1 = F12 + F13 + F14 + F15 F12 F14 F15 F13 F1 qa qb 1 4πε0 Fa (ra) = ∑ b=1 b≠a N ra - rb ra - rb│ │3 In the present example, a = 1 and b = 2 to 5. If the force is to be found on 2nd charge, then a = 2 and b = 1 and 3 to 5. Superposition principle holds good for electric field also. Note: The interactions must be on the charge which is to be studied due to other charges. The charge on which the influence due to other charges is to be found is assumed to be floating charge and others are rigidly fixed. For eg. 1st charge (floating) is repelled away by q2 and q4 and attracted towards q3 and q5. The interactions between the other charges (among themselves) must be ignored. i.e. F23, F24, F25, F34, F35 and F45 are ignored. Electric Lines of Force: An electric line of force is an imaginary straight or curved path along which a unit positive charge is supposed to move when free to do so in an electric field. Electric lines of force do not physically exist but they represent real situations. Electric Lines of Force E E 4. Electric Lines of Force due to a Uniform Field: + + + + - - - - E Properties of Electric Lines of Force or Field Lines: 1. The electric lines of force are imaginary lines. 2. A unit positive charge placed in the electric field tends to follow a path along the field line if it is free to do so. 3. The electric lines of force emanate from a positive charge and terminate on a negative charge. 4. The tangent to an electric field line at any point gives the direction of the electric field at that point. 5. Two electric lines of force can never cross each other. If they do, then at the point of intersection, there will be two tangents. It means there are two values of the electric field at that point, which is not possible. Further, electric field being a vector quantity, there can be only one resultant field at the given point, represented by one tangent at the given point for the given line of force. E1 E2 E +1 C E. P NOT POSSIBLE 6. Electric lines of force are closer (crowded) where the electric field is stronger and the lines spread out where the electric field is weaker. 7. Electric lines of force are perpendicular to the surface of a positively or negatively charged body. Q > q qQ 8. Electric lines of force contract lengthwise to represent attraction between two unlike charges. 9. Electric lines of force exert lateral (sideways) pressure to represent repulsion between two like charges. 10.The number of lines per unit cross – sectional area perpendicular to the field lines (i.e. density of lines of force) is directly proportional to the magnitude of the intensity of electric field in that region. 11. Electric lines of force do not pass through a conductor. Hence, the interior of the conductor is free from the influence of the electric field. Solid or hollow conductor No Field + + + + + + + - - - - - - - + + + + E E - - - - α E ∆N ∆A 12. Electric lines of force can pass through an insulator. (Electrostatic Shielding) θθ ll y θ θ + q- q p A B EB EA EQ Q Resultant electric field intensity at the point Q is EQ = EA + EB The vectors EA and EB are acting at an angle 2θ. ii) At a point on the equatorial line: q ( x2 + l2 ) 1 4πε0 EA = i q1 4πε0 EB = i ( x2 + l2 ) EB EA EQ θ θEA cos θ EB cos θ EB sin θ EA sin θ The vectors EA sin θ and EB sin θ are opposite to each other and hence cancel out. The vectors EA cos θ and EB cos θ are acting along the same direction and hence add up. EQ = EA cos θ + EB cos θ EQ = q2 4πε0 ( x2 + l2 ) l ( x2 + l2 )½ 1 EQ = 4πε0 q . 2l ( x2 + l2 )3/2 EQ = 1 4πε0 p ( x2 + l2 )3/2 Q O EQ = 1 4πε0 p ( x2 + l2 )3/2 (- i ) If l << y, then EQ ≈ p 4πε0 y3 The direction of electric field intensity at a point on the equatorial line due to a dipole is parallel and opposite to the direction of the dipole moment. If the observation point is far away or when the dipole is very short, then the electric field intensity at a point on the axial line is double the electric field intensity at a point on the equatorial line. i.e. If l << x and l << y, then EP = 2 EQ Torque on an Electric Dipole in a Uniform Electric Field: The forces of magnitude pE act opposite to each other and hence net force acting on the dipole due to external uniform electric field is zero. So, there is no translational motion of the dipole. θ However the forces are along different lines of action and constitute a couple. Hence the dipole will rotate and experience torque. Torque = Electric Force x distance θ t = q E (2l sin θ) = p E sin θ q E q E + q - q E 2l t p E Direction of Torque is perpendicular and into the plane containing p and E. SI unit of torque is newton metre (Nm). Case i: If θ = 0°, then t = 0. Case ii: If θ = 90°, then t = pE (maximum value). Case iii: If θ = 180°, then t = 0. t = p x E p Line Integral of Electric Field (Work Done by Electric Field): Negative Line Integral of Electric Field represents the work done by the electric field on a unit positive charge in moving it from one point to another in the electric field. +q0 O Z Y X + q F A rA B rB r WAB = dW = - E . dl A B Let q0 be the test charge in place of the unit positive charge. The force F = +q0E acts on the test charge due to the source charge +q. It is radially outward and tends to accelerate the test charge. To prevent this acceleration, equal and opposite force –q0E has to be applied on the test charge. Total work done by the electric field on the test charge in moving it from A to B in the electric field is = 11qq0 4πε0 ][ - rB rA WAB = dW = - E . dl A B 1. The equation shows that the work done in moving a test charge q0 from point A to another point B along any path AB in an electric field due to +q charge depends only on the positions of these points and is independent of the actual path followed between A and B. 2. That is, the line integral of electric field is path independent. 3. Therefore, electric field is ‘conservative field’. 4. Line integral of electric field over a closed path is zero. This is another condition satisfied by conservative field. Note: Line integral of only static electric field is independent of the path followed. However, line integral of the field due to a moving charge is not independent of the path because the field varies with time. E . dl = 0 A B = 11qq0 4πε0 ][ - rB rA WAB = dW = - E . dl A B Electric potential is a physical quantity which determines the flow of charges from one body to another. It is a physical quantity that determines the degree of electrification of a body. Electric Potential at a point in the electric field is defined as the work done in moving (without any acceleration) a unit positive charge from infinity to that point against the electrostatic force irrespective of the path followed. Electric Potential: = 11qq0 4πε0 ][ - rB rA WAB = - E . dl A B = 11q 4πε0 ][ - rB rA WAB q0 According to definition, rA = ∞ and rB = r (where r is the distance from the source charge and the point of consideration) = q 4πε0 r W∞B q0 = V V = W∞B q0 SI unit of electric potential is volt (V) or J C-1 or Nm C-1. Electric potential at a point is one volt if one joule of work is done in moving one coulomb charge from infinity to that point in the electric field. or Electric Potential due to a Group of Point Charges: +1 C q2 qn q4 P q1 r1 r2 r3r4 rnVP = V1 + V2 + V3 + V4 + …………+ Vn │ │ 1 4πε0 V = ∑ i=1 n qi r - ri ( in terms of position vector ) The net electrostatic potential at a point in the electric field due to a group of charges is the algebraic sum of their individual potentials at that point. 1. Electric potential at a point due to a charge is not affected by the presence of other charges. 2. Potential, V α 1 / r whereas Coulomb’s force F α 1 / r2. 3. Potential is a scalar whereas Force is a vector. 4. Although V is called the potential at a point, it is actually equal to the potential difference between the points r and ∞. 1 4πε0 V = ∑ i=1 n qi ri q3 Electric Potential due to an Electric Dipole: ll x P+ q- q p A B +1 C q VP q+ = 4πε0 (x – l) 1 VP = VP q+ + VP q- VP q- = 4πε0 (x + l) 1 - q VP = 4πε0 q (x – l) 1[ - (x + l) 1 ] VP = 1 4πε0 q . 2l (x2 – l2) VP = 1 4πε0 p (x2 – l2) i) At a point on the axial line: O + q- q p A Bθθ y O ii) At a point on the equatorial line: q VQ q+ = 4πε0 BQ 1 VQ = VP q+ + VP q- VQ q- = 4πε0 AQ 1 - q VQ = 4πε0 q BQ 1[ - AQ 1 ] VQ = 0 BQ = AQ The net electrostatic potential at a point in the electric field due to an electric dipole at any point on the equatorial line is zero. Q ll 4. Two equipotential surfaces can not intersect. If two equipotential surfaces intersect, then at the points of intersection, there will be two values of the electric potential which is not possible. (Refer to properties of electric lines of force) 3. Equipotential surfaces indicate regions of strong or weak electric fields. dV dr E = - Electric field is defined as the negative potential gradient. or dV E dr = - Since dV is constant on equipotential surface, so E 1 dr α If E is strong (large), dr will be small, i.e. the separation of equipotential surfaces will be smaller (i.e. equipotential surfaces are crowded) and vice versa. Note: Electric potential is a scalar quantity whereas potential gradient is a vector quantity. The negative sign of potential gradient shows that the rate of change of potential with distance is always against the electric field intensity. Electrostatic Potential Energy: The work done in moving a charge q from infinity to a point in the field against the electric force is called electrostatic potential energy. W = q V i) Electrostatic Potential Energy of a Two Charges System: O Z Y X A (q1) r1 B (q2) r2 - r1r2 U = q1q2 4πε0 │ │- r1 r2 1 or U = q1q2 4πε0 r12 1 ii) Electrostatic Potential Energy of a Three Charges System: O Z Y X A (q1) r1 B (q2) r2 or C (q3) r3 - r1r3 - r2r3 U = q1q2 4πε0 │ │- r1 r2 1 + q1q3 4πε0 │ │- r1 r3 1 + q2q3 4πε0 │ │- r2 r3 1 U = q1q2 4πε0 r12 1 [ q1q3 r31 q2q3 r32 + + ] iii) Electrostatic Potential Energy of an n - Charges System: 1 4πε0 U = ∑ j=1 i≠j n qi qj rj - ri│ │ ∑ i=1 n[ 2 1 ] - r1r2 Gauss’s Theorem: The surface integral of the electric field intensity over any closed hypothetical surface (called Gaussian surface) in free space is equal to 1 / ε0 times the net charge enclosed within the surface. E . dS = S ΦE = 1 ε0 ∑ i=1 n qi Proof of Gauss’s Theorem for Spherically Symmetric Surfaces: E . dSdΦ = r2 1 4πε0 = q r . dS n dΦ = r2 1 4πε0 q dS r n. Here, = 1 x 1 cos 0°= 1r n. dΦ = r2 1 4πε0 q dS S ΦE = dΦ r2 1 4πε0 q = 4π r2 ε0 q =dS S r2 1 4πε0 q = O • r r dS E +q Proof of Gauss’s Theorem for a Closed Surface of any Shape: EE . dSdΦ = r2 1 4πε0 = q r . dS n dΦ = r2 1 4πε0 q dS r n. Here, = 1 x 1 cos θ = cos θ r n. dΦ = r2 q 4πε0 dS cos θ S ΦE = dΦ q 4πε0 = 4π ε0 q =dΩ q 4πε0 S = dΩ r θ dS n r +q • O • r r dS E +q Deduction of Coulomb’s Law from Gauss’s Theorem: From Gauss’s law, E . dS = S ΦE = q ε0 E dS = S ΦE = q ε0 or dS = S ΦE = q ε0 E E = q 4πε0 r2 E x 4π r2 q ε0 = If a charge q0 is placed at a point where E is calculated, then Since E and dS are in the same direction, which is Coulomb’s Law. or F = qq0 4πε0 r2 E dS C B A E E dS dS r l 2. Electric Field Intensity due to an Infinitely Long, Thin Plane Sheet of Charge: From Gauss’s law, σ E . dS = S ΦE = q ε0 E . dS = S E . dS + A E . dS + B E . dS C E . dS = S E dS cos 0°+ A B E dS cos 0°+ C E dS cos 90°= 2E dS = 2E x π r2 TIP: The field lines remain straight, parallel and uniformly spaced. (where σ is the surface charge density) or E = 2 ε0 σ In vector form, E (l) = 2 ε0 σ l The direction of the electric field intensity is normal to the plane and away from the positive charge distribution. For negative charge distribution, it will be towards the plane. Note: The electric field intensity is independent of the size of the Gaussian surface constructed. It neither depends on the distance of point of consideration nor the radius of the cylindrical surface. q ε0 = σ π r2 ε0 2 E x π r2 = σ π r2 ε0 If the plane sheet is thick, then the charge distribution will be available on both the sides. So, the charge enclosed within the Gaussian surface will be twice as before. Therefore, the field will be twice. E = ε0 σ 3. Electric Field Intensity due to Two Parallel, Infinitely Long, Thin Plane Sheet of Charge: σ1 σ2 E1E1E1 E2E2E2 E E E Region I Region II Region III E = E1 + E2 E = 2 ε0 σ1 + σ2 E = E1 - E2 E = 2 ε0 σ1 - σ2 E = E1 + E2 E = 2 ε0 σ1 + σ2 σ1 > σ2( ) Case 1: σ1 > σ2 4. Electric Field Intensity due to a Uniformed Charged This Spherical Shell: dS E q r R •PFrom Gauss’s law, E . dS = S ΦE = q ε0 E dS = S ΦE = q ε0 or dS = S ΦE = q ε0 E E x 4π r2 q ε0 = Since E and dS are in the same direction, or E = q 4πε0 r2 i) At a point P outside the shell: Since q = σ x 4π R2, E = ε0 r2 σ R2 Electric field due to a uniformly charged thin spherical shell at a point outside the shell is such as if the whole charge were concentrated at the centre of the shell. HOLLOW ……… Gaussian Surface O • dS EFrom Gauss’s law, E . dS = S ΦE = q ε0 E dS = S ΦE = q ε0 or dS = S ΦE = q ε0 E E x 4π R2 q ε0 = Since E and dS are in the same direction, or E = q 4πε0 R2 ii) At a point A on the surface of the shell: Electric field due to a uniformly charged thin spherical shell at a point on the surface of the shell is maximum. Since q = σ x 4π R2, E = ε0 σ q R HOLLOW O • • A dS E r’ Oq R HOLLOW • B • From Gauss’s law, E . dS = S ΦE = q ε0 E dS = S ΦE = q ε0 or dS = S ΦE = q ε0 E E x 4π r’2 q ε0 = Since E and dS are in the same direction, or E = 0 4πε0 r’2 iii) At a point B inside the shell: This property E = 0 inside a cavity is used for electrostatic shielding. (since q = 0 inside the Gaussian surface) E = 0 r E R O Emax 3. Net charge in the interior of a conductor is zero. The charges are temporarily separated. The total charge of the system is zero. E . dS = S ΦE = q ε0 Since E = 0 in the interior of the conductor, therefore q = 0. 4. Charge always resides on the surface of a conductor. Suppose a conductor is given some excess charge q. Construct a Gaussian surface just inside the conductor. Since E = 0 in the interior of the conductor, therefore q = 0 inside the conductor. q = 0 q q 5. Electric potential is constant for the entire conductor. dV = - E . dr Since E = 0 in the interior of the conductor, therefore dV = 0. i.e. V = constant 6. Surface charge distribution may be different at different points. σ = q S σmax σmin Every conductor is an equipotential volume (three- dimensional) rather than just an equipotential surface (two- dimensional). Electrical Capacitance: The measure of the ability of a conductor to store charges is known as capacitance or capacity (old name). q α V or q = C V or C = q V If V = 1 volt, then C = q Capacitance of a conductor is defined as the charge required to raise its potential through one unit. SI Unit of capacitance is ‘farad’ (F). Symbol of capacitance: Capacitance is said to be 1 farad when 1 coulomb of charge raises the potential of conductor by 1 volt. Since 1 coulomb is the big amount of charge, the capacitance will be usually in the range of milli farad, micro farad, nano farad or pico farad. Capacitance of an Isolated Spherical Conductor: O • r +q Let a charge q be given to the sphere which is assumed to be concentrated at the centre. Potential at any point on the surface is V = q 4πε0 r C = q V C = 4πε0 r 1. Capacitance of a spherical conductor is directly proportional to its radius. 2. The above equation is true for conducting spheres, hollow or solid. 3. IF the sphere is in a medium, then C = 4πε0εr r. 4. Capacitance of the earth is 711 µF. Series Combination of Capacitors: V1 V2 V3 V C1 C2 C3In series combination, i) Charge is same in each capacitor ii) Potential is distributed in inverse proportion to capacitances i.e. V = V1 + V2 + V3 But q V1 = C1 V2 = C2 q V3 = C3 q , and q V = C , (where C is the equivalent capacitance or effective capacitance or net capacitance or total capacitance) q = C1 + C2 q + C3 qq C or The reciprocal of the effective capacitance is the sum of the reciprocals of the individual capacitances. Note: The effective capacitance in series combination is less than the least of all the individual capacitances. q q q ∑ i=1 n 1 Ci 1 C = 1 = C1 + C2 1 + C3 11 C Parallel Combination of Capacitors: In parallel combination, i) Potential is same across each capacitor ii) Charge is distributed in direct proportion to capacitances i.e. q = q1 + q2 + q3 But , and, (where C is the equivalent capacitance) or The effective capacitance is the sum of the individual capacitances. Note: The effective capacitance in parallel combination is larger than the largest of all the individual capacitances. q1 = C1 V q2 = C2 V q3 = C3 V q = C V C V = C1V + C2 V + C3 V ∑ i=1 n CiC =C = C1 + C2 + C3 V q1 C1 C2 C3 V V V q2 q3 Energy Stored in a Capacitor: V The process of charging a capacitor is equivalent to transferring charges from one plate to the other of the capacitor. The moment charging starts, there is a potential difference between the plates. Therefore, to transfer charges against the potential difference some work is to be done. This work is stored as electrostatic potential energy in the capacitor. If dq be the charge transferred against the potential difference V, then work done is dU = dW = V dq q = C dq The total work done ( energy) to transfer charge q is U = 0 q q C dq U = q2 C 1 2 U = 1 2 C V2 U = 1 2 q Vor or or The total energy before sharing is Ui = 1 2 C1 V1 2 1 2 C2 V2 2+ The total energy after sharing is Uf = 1 2 (C1 + C2) V2 Ui– Uf = C1 C2 (V1 – V2)2 2 (C1 + C2) Ui – Uf > 0 or Ui > Uf Therefore, there is some loss of energy when two charged capacitors are connected together. The loss of energy appears as heat and the wire connecting the two capacitors may become hot. Polar Molecules: A molecule in which the centre of positive charges does not coincide with the centre of negative charges is called a polar molecule. Polar molecule does not have symmetrical shape. Eg. H Cl, H2 O, N H3, C O2, alcohol, etc. O H H 105° Effect of Electric Field on Polar Molecules: E = 0 E p = 0 p In the absence of external electric field, the permanent dipoles of the molecules orient in random directions and hence the net dipole moment is zero. When electric field is applied, the dipoles orient themselves in a regular fashion and hence dipole moment is induced. Complete allignment is not possible due to thermal agitation. p Non - polar Molecules: A molecule in which the centre of positive charges coincides with the centre of negative charges is called a non-polar molecule. Non-polar molecule has symmetrical shape. Eg. N2 , C H4, O2, C6 H6, etc. Effect of Electric Field on Non-polar Molecules: E = 0 E p = 0 p In the absence of external electric field, the effective positive and negative centres coincide and hence dipole is not formed. When electric field is applied, the positive charges are pushed in the direction of electric field and the electrons are pulled in the direction opposite to the electric field. Due to separation of effective centres of positive and negative charges, dipole is formed. Capacitance of Parallel Plate Capacitor with Dielectric Slab: EpE0 EN = E0 - Ep dt V = E0 (d – t) + EN t EN K = E0 or EN = K E0 V = E0 (d – t) + K E0 t V = E0 [ (d – t) + K t ] But E0 = ε0 σ = ε0 qA and C = q V C = A ε0 [ (d – t) + K t ] or C = A ε0 d [1 – K t ]d t (1 - ) or C = C0 [1 – K t ]d t (1 - ) C > C0. i.e. Capacitance increases with introduction of dielectric slab. If the dielectric slab occupies the whole space between the plates, i.e. t = d, then WITH DIELECTRIC SLAB Increases (K U0)Remains the same Energy stored Remains the sameDecreasesPotential Difference Remains the sameDecreases EN = E0 – Ep Electric Field Increases (K C0)Increases (K C0)Capacitance Increases (K C0 V0)Remains the sameCharge With Battery connected With Battery disconnected Physcial Quantity C0 K = C C = K C0 Dielectric Constant Van de Graaff Generator: T D C2 C1 P1 P2 M S I S HVR S – Large Copper sphere C1, C2 – Combs with sharp points P1, P2 – Pulleys to run belt HVR – High Voltage Rectifier M – Motor IS – Insulating Stand D – Gas Discharge Tube T - Target Working: Let the positive terminal of the High Voltage Rectifier (HVR) is connected to the comb (C1). Due to action of points, electric wind is caused and the positive charges are sprayed on to the belt (silk or rubber). The belt made ascending by electric motor (EM) and pulley (P1) carries these charges in the upward direction. The comb (C2) is induced with the negative charges which are carried by conduction to inner surface of the collecting sphere (dome) S through a metallic wire which in turn induces positive charges on the outer surface of the dome. The comb (C2) being negatively charged causes electric wind by spraying negative charges due to action of points which neutralize the positive charges on the belt. Therefore the belt does not carry any charge back while descending. (Thus the principle of conservation of charge is obeyed.) Contd.. The process continues for a longer time to store more and more charges on the sphere and the potential of the sphere increases considerably. When the charge on the sphere is very high, the leakage of charges due to ionization of surrounding air also increases. Maximum potential occurs when the rate of charge carried in by the belt is equal to the rate at which charge leaks from the shell due to ionization of air. Now, if the positively charged particles which are to be accelerated are kept at the top of the tube T, they get accelerated due to difference in potential (the lower end of the tube is connected to the earth and hence at the lower potential) and are made to hit the target for causing nuclear reactions, etc. Uses: Van de Graaff Generator is used to produce very high potential difference (of the order of several million volts) for accelerating charged particles. The beam of accelerated charged particles are used to trigger nuclear reactions. The beam is used to break atoms for various experiments in Physics. In medicine, such beams are used to treat cancer. It is used for research purposes.