Electrostatic potential and capacitance, Exercises of Physics

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ELECTROSTATICS Vedanti ELECTRIC CHARGE 1.1 Definition Charge is the property associated with matter due to which it produces and experiences electrical and magnetic effects. 12 Type There exists two types of charges in nature @ Positive charge Gd = Negative charge Charges with the same electrical sign repel each other, and charges with opposite electrical sign attract each other. ——— 1.3 Unit and dimensional formula S.L unit of charge is coulomb (C), (ime =107C, Inc =10°C, Inc= 19°C). C.GS. unit of charge is 2.5.4. 1C=3 x10 esu Dimensional formula [Q]=[AT]. 1.4 Point Charge Whose spatial size is negligible as compared to other distances. 1.5 Properties of charge @ Chargeis a Scalar Quantity : Charges can be added or subtracted algebrically. di) Chargeis transferable : Ifa charged body is put in contact with an uncharged body, uncharged body becomes charged due to transfer of electrons from one body to the other. (ii) @y) wi) (vii) (viii) Charge is always associated with mass, i¢., charge can not exist without mass though mass can exist without charge. Charge is conserved : Charge canneither be created nor be destroyed. Invariance of charge : The numerical value of an elementary charge is independent of velocity. Charge produces electric fidd and magnetic field: A charged particle at rest produces only electric field in the space surrounding it. However, if the charged particle is in unaccelerated motion it produces both electric and magnetic fields. And if the motion of charged particle is accelerated it not only produces electric and magnetic fields but also radiates energy in the space surrounding the charge in the form of electromagnetic waves. Chargeresides on the surface of conductor : Charge resides on the outer surface of a conductor because like charges repel and try to get as far away as possible from one another and stay at the farthest distance from each other which is outer surface of the conductor. This is why a solid and hollow conducting sphere of same outer radius will hold maximum equal charge and a soap bubble expands on charging. Quantization of charge : When a physical quantity can have only discrete values rather than any value, the quantity is said to be quantised. The smallest charge that can exist in nature is the charge of an electron. If the charge of an electron (-L.6x10"?c) is taken as elementary unit ie. quanta of charge the charge on any body will be some integral multiple ofeie., Q=+newith n =0, Charge ona body can never be 0.5 e, +17.2 eor +10° e etc. 1.6 Comparison of Charge and Mass We are familiar with role of mass in gravitation, and we have just studied some features of electric charge. We can compare the two as shown below Charge Mass 1. Flecirie charge canbe | 1. Mass ofa body isa positive, negative or sero, positive quantity + 2. Charge carried by a body Ww Mass of a body increases does not depend upon with ils velocity as My velocity of the body m— where © 1s velocity of light in vaccum, m is the mass of the velocity v and m, is rest mass of the body. wo Charge is quantized, 3. The quantization of mass is yet to be established. 4 Electric charge isalways | 4. Mass is not conserved as conserved. it can be changed into energy and vice-versa. wn wa Force between charges The gravitational force can be allractive or between Lwo masses is repulsive, according as always aliractive charges are unlike or Tike charges 1.7 Methods of Charging Abody can be charged by following methods {By friction : Tn friction when wo bodies are rubbed together, clestrons are transferred from one body to the other. As a result of this one body becomes positively charged white the other newatively charged, cz, when a glass rod is rubbed with silk, the red becomes positively charged while the silk becomes negatively charged. However, ebonite onrubbing with wool becomes negatively charged making the wool positively charged. Clouds also become charged by friction. In charging by friction in accordance with conservation of charge, both positive and negative charges in equal amounts appear simmultancously duc to transfer of electrons from one body to the other. (ii) By electrostatic induction : If a charged body is brought near an uncharged body, the charged body will attract opposite charge and repel similar charge present in the uncharged body. As a result of this one side of neutral bedy (closer to charged bedy} becomes oppositely charged while the other is similarly charged. This process is called electrostatic induction. Hin, Inducting body neither gains nor loses charge. (iii) | Charging by conduction : ‘lake two conductors, one charged and other uncharged. Bring the conductors in contact with each other. The charge (whether -ve or + ve) under its own repulsion will spread over both the conductors. Thus the conductors will be charged with the same sign This is called as charging by conduction (through contact). 2.1.1 Effect of units @ InC.GS. forair f= 1, F= 222 Dyne E N-m? Ca 1 ) InS.L forair k => = 9x10" Tey 1 =. 12> Newton (1 Newton=10°Dyne) 4m or Sa * €, = Absolute permittivity of air or free space =sesxio? © _(_Famd) N-m’ m Dimension is [aot Tta? | * €, Relates with absolute magnetic permeability (up) and velocity of light (c) according to the following relation ¢ = a Vbo% 2.12 Effect ofmedium (a) When a dielectric medium is completely filled in between charges rearrangement of the charges inside the dielectric medium takes place and the force between the same two charges decreases by a factor of K known as dielectric constant, K is also called relative permittivity € of the medium (relative means with respect to free space). Hence in the presence of medium po ufie._ 1 = K 4ne,K” fr? Here 2,K = e)2, = (permittivity of medium) Medium K ‘Vacuum / air 1 Water 80 Mica 6 Glass 5410 Metal a) 2.2 Vector form of coulomb’s law It is helpful to adopt a convention for subscript notation. F,, =force on 1 dueto 2 F,, = force on 2 due to 1 Suppose the position vectors of two charges q, and q, are 7, and 1, then, electric force on charge q, due to charge q, is, B 1442 is 2 Ane, la-#P ( Similarly, electric force on q, due to charge q, is = 1 4% @ = B= 12g 21 an [BF (®-i) Force isa vector, so in vector form the Coulomb’s law is written as = 1 * i 42 = M2 Ane, 1? where ij, is aunit vector directed toward q, from q,. tute Fo hat Z 1 442 5 1 424i (_¢ R= Sa = are, 12 4ne, r? Cha) 1 GoM. __F = ee ae Ane, we a a te 2 hi LL from d . trom to to Remember convention for 7. Here q, and q, are to be substituted with sign. Position vector of charges q, and q, are ¥ =xi+yitzk and R= X)it+y,j+ Zk respectively. Where (x,, y,, Z,) and (x, Y, Z) are the co-ordinates of charges q, and q,. 2.3 Principle of superposition According to the principle of super position, total force acting on a given charge due to number of charges is the vector sum of the individual forces acting on that charge due to all the charges. Consider number of charge Q,, Q,, Q,..-are applying force ona charge QO Net force on Q will be The magnitude of the resultant of two electric force is given by nN F} +2KF, cos@ and the force direction is given by FE sin E+E, cosé tan @ 3. ELECTRIC FIELD A positive charge or a negative charge is said to create its field around itself Thus space around a charge in which another charged particle experiences a force is said to have electrical field in it 3.1 Electric field intensity (E) The electric field intensity at any point is defined as the force experienced by a unit positive charge placed at that point E= 2 | Where q, > 0so that presence of this charge may not affect the source charge Q and its electric field is not changed, therefore expression for electric field intensity can be better written as E= lim a= 47° do The number of Field Lines passing through perpendicular unit area will be proportional to the magnitude of Electric Field there. Tangent toa Field line at any point gives the direction of Electric Field at that point. This will be the instantaneous path charge will take ifkept there. Two or more field lines can never intersect each other. [they cannot have multiple directions] Uniform field lines are straight, parallel & uniformly placed. Field lines cannot forma loop. Let a charge particle of mass m and charge Q be initially at rest in an electric field of strength Z s ———_———_>F +Q Q@—> F-o —_ _*=s F-QF +—O -Q —_—S a Fig, (A) Fig. B) 8 Electric field lines originate & terminate perpendicular to the surface of the conductor. Electric field lines do not exist inside a conductor. 47+] +]+1+F i (A) (B) 9 Field lines always flow from higher potential to lower potential. 10. Ifinaregion electric field is absent, there will be no field lines. 3.7 Motion of Charged Particle in an Electric Field (a) = When charged particle initially at rest is placed in the uniform field : @ di) Force and acceleration : The force experienced by the charged particle is F = QE. Positive charge experiences force in the direction of electric field while negative charge experiences force in the direction opposite to the field. [Fig. (A)] Acceleration produced by this force is a= Since the field E in constant the acceleration is constant, thus motion of the particle is uniformly accelerated. Velocity : Suppose at point A particle is at rest and in time 4, itreaches the point B [Fig. (B)] ¥ =Potential difference between.A and B; S=Separation between A and B (a By using E v=utat, v=0+Q—t, m yt m (b) By using v? =u? 42a8, v2 = 042% eB xs m Gii) Momentum: Momentump=my, p=mx 2 - ort m dy) Kinetic energy : Kinetic energy gained by the particle in time tis (QEty = OE? m om b) When a charged particle enters with an initial velocity at right angle to the uniform field When charged particle enters perpendicularly in an electric field, it describe a parabolic path as shown @ Equation of trajectory : Throughout the motion particle has uniform velocity along x-axis and horizontal displacement (x) is given by the equation x=ut Since the motion of the particle is accelerated along y-axis, we will use equation of motion for uniform acceleration to 1 determine displacement y. From S = ut+ 5 at? We have u = 0 (along y-axis) so y = at? ie., displacement along y-axis will increase rapidly with time Gincey « t?) From displacement along x-axis t =x/u 1/(QE)(x) ; So Y= >|—]||—] ; this is the equation of parabola 24m )\u which shows y oc x? Gi) ‘Velocity at any instant : At any instanté, v,, =u and If is the angle made by v with x-axis than v. tanp = M5: _ GE v, mu 4, ELECTRIC POTENTIAL ENERGY 4.1 Potential energy of 2 charges system It is always change in potential energy that is designed as AU =- W, =-W, conservative fence Coulord force Potential energy is defined of a system of charges in a particular configuration. Consider a system of two charges q, and q,. Suppose, the charge q, is fixed and the charge q, is taken from a pointA to moved The electric force on the charge q, is F = 9192 _ Ane, The total work done as the charge q, moves from B to C is ‘ we f itt, dr= a%(2-1) 2 Anegr 4ne, qt) Q >Vp = q Ane,r Aner (Veo is taken as 0) The electric potential due to a system of charges may be obtained by finding potentials due to the individual charges using equation and then adding them. Thus, = oS. Ane, q Electric potential is a scalar quantity, hence sign of charges is to taken in expression it is denoted by V 5.2 Unit and dimensional formula Joule 8.1 unit— = volt Coulomb — [VI=[MLAT 2A] 5.3 Types of electric potential According to the nature of charge potential is of two types (i) Positive potential : Due to positive charge. (ii) Negative potential : Due to negative charge. Saft *% =~ At the centre of two equal and opposite charge V=O0butE#0. % ~~ At the centre of the line joining two equal and similar charge V #0, E=0. * Ifleftfreetomove, Positive charge will always move from higher to lower potential points. Negative charge will always move from lower to higher potential points. (Because this motion will decrease potential energy of a system) 6, RELATION BETWEEN ELECTRIC FIELD & POTENTIAI In an electric field rate of change of potential with distance is known as potential gradient. It is a vector quantity and it’s direction is opposite to that of electric field. Potential gradient relates with electric field according to the following relation = — ©; This relation gives another unit of sis, ; volt electric field is m aa In the above rleation negative sign indicates that in the direction of electric field potential decreases. In space around a charge distribution we can also write E=E,i+E,j+E,k where E,, HES aul and E, a lV dk’ ody dz Suppose, B and C are three points in an uniform electric field as shown in figure. @ Potential difference between point 4 and B is Ba LY Va-Va=-lE. at A Since displacement is inthe direction of electric field, hence a= B B So, V,-V, =-]E. dr cos0=—JE. dr=—Ed A A Equipotential Surface or Lines a (e2) ) @ 6) Ifevery point of a surface is at same potential, then it is said to be an equipotential surface dark for a given charge distribution, locus of all points having same potential is called “equipotential surface” regarding equipotential surface following points should keep in mind The direction of electric field is perpendicular to the equipotential surfaces or lines. The equipotential surfaces produced by a point charge or a spherically charge distribution are a family of concentric spheres. §<—Tquipotential surface For auniform electric field, the equip otential surfaces area family of plane perpendicular to the field lines. A metallic surface of any shape is an equipotential surface e.g. When a charge is given to a metallic surface, it distributes itself in a manner such that its every point comes at same potential even if the object is of irregular shape and has sharp points on it Metallic charged sphere Charged metallic body of irregular shape © Equipotential surfaces can never cross each other. It is a common misconception that the path traced by a positive test charge is a field line but actually the path traced by a unit positive test charge represents a field full line only when it moves along a straight line. 7. ELECTRIC DIPOLE 7.1 General information System of two equal and opposite charges separated by a small fixed distance is called a dipole. “Or u-—— 2 ——5 B f)}. O q Dipole axis et @ Gi) did) dy) ote. * * Dipole axis : Line joining negative charge to positive charge of a dipole is called its axis. It may also be termed as its longitudinal axis. Equatorial axis : Perpendicular bisector of the dipole is called its equatorial or transverse axis as it is perpendicular to length. Dipole length : The distance between two charges is known as dipole length (d) Dipole moment : It is a quantity which gives information ab out the strength of dipole. It is a vector quantity and is directed from negative charge to positive charge along the axis. It is denoted as p and is defined as the product of the magnitude of either of the charge and the dipole length. ie. p= ald) Its S.L unit is coulomb-metre or Debye (1 Debye = 3.3 x 10 Cx m) and its dimensions are AP Z'7'A!. 4 A region surrounding a stationary electric dipole has electric field only. When a dielectric is placed in an electric field, its atoms or molecules are considered as tiny dipoles. The directions of. Ey and E_, are as shown in fig. (b). Clearly, the components normal to the dipole axis cancel away. The components along the dipole axis add up. The total electric field is opposite to P. We have E=-(,,+E_,) cos@ p _ 2qa ras Aneo(t? +a ee At large distances (r >> a), this reduces to 2qa_ a. P @>>a) ii) E=4 From Egg. (i) and (ii), it is clear that the dipole field at large distances does not involve q and a separately ; it depends on the product qa. This suggests the definition of dipole is defined by P=qx2ap that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, —q) and the direction is along the line from -q to q. In terms ofp, the electric field of a dipole at large distances takes simple forms : Ata point on the dipole axis E=——, EB (r >> a) 4negr r 0: At a point on the equatorial plane kp 3 3 E=-—? 4nzor ie (r >> a) 7.3 Electric Dipole in uniform electric field @ Force and Torque : Ifa dipole is placed ina uniform field such that dipole @e. p ) makes an angle 6 with direction of field then two equal and opposite force acting on dipole constitute a couple whose tendency is to rotate the dipole hence a torque is developed in it and dipole tries to align it self in the direction of field. Consider an electric dipole in placed ina uniform electric field such that dipole (ie. p ) makes an angle 6 with the direction of electric field as shown Gi) (ii) @ Net force on electric dipole F,,, =0 ) vo. U=pE sin@ (¢=pxE) Work : From the above discussion it is clear that in an uniform electric field dipole tries to align itself in the direction of electric field (i.e. equilibrium position). To change it’s angular position some work has to be done. Suppose an electric dipole is kept in an uniform electric field by making an angle @, with the field, ifitis again turn so that it makes an angle @, with the field, work done in this process is given by the formula W =pE(cos®, —cos6, ) Potential energy : In case of a dipole (in a uniform field), potential energy of dipole is defined as work done in rotating a dipole from a direction perpendicular to the field to the given direction ie. if 6, = 90° and 8, =6 then | W=AU=U,-U,,,=—pEcos 0 =>U,=-pEcos@ +: [U(90°)=0] or 8. ELECTRIC FIELD 8.1 Continuous charge distributions Diagram Data Graph K SExExERESEEEI ped Fp = —KAL___kQ LQa <7 d(L+d) d(L+4) —-— Or ——< toeee+ettst}+—* Sule In the above formula if Q, = Q,, neutral point lies at the centre so remember that resultant field at the midpoint of two equal and like charges is zero. (b) = At anexternal point along the line joining two unlike charges (Duetoa system of twounlike point charge) : Suppose two unlike charge Q, and Q, separated by a distance.x from each other. — f—1 a os Here neutral point lies outside the line joining two unlike charges and also it lies nearer to charge which is smaller in magnitude. Tf |Q,| <|Q,| then neutral point will be obtained on the side of Q,, suppose it is at a distance / from Q, Hence at neutral point ; KQl_ kal Sf t ) f «tay (Q x+é x (¥QI@l- Short trick : £= a In the above discussion if |Q,| = |Q,| neutral point will beat infinity. 8.3 Equilibrium of Charge (a) _— Definition: Acharge is said to be in equilibrium, ifnet force acting on it is zero. A system of charges is said to be in equilibrium if each charge is in equilibrium. (&) Type of equilibrium : Equilibrium can be divided in following type: @ Stable equilibrium : After displacing a charged particle from it’s equilibrium position, if it returns back then it is said to be in stable equilibrium. If Uis the potential energy then in case of stable equilibrium U is minimum. Gi) Unstable equilibrium : After displacing a charged particle from it’s equilibrium position, if it never returns back then it is said to be in unstable equilibrium and in unstable equilibrium, U is maximum ii) Neutral equilibrium : After displacing a charged particle from it’s equilibrium position if it neither comes back, nor moves away but remains in the position in which it was kept it is said to be in neutral equilibrium and in neutral equilibrium, Uis constant. (Oo Different cases of equilibrium of charge Suppose three similar charge Q,, q and Q, are placed along a straight line as shown below Case-1: cs Charge qwill be in equilibrium if|F,|=|F,ée., Qe [2] y Q Le This is the condition of equilibrium of charge g. After following the guidelines we can say that charge gis in stable equilibrium and this system is not in equilibrium. uo x x)= and x, = 1447 * 144Q)/Q; e.g. if two charges +4 uC and +16 uC are separated by a distance of 30 cm from each other then for equilibrium a third charge should be placed between them at a distance 30 Xx; =———— = 10 em orx,=20em 1evi6/4 4 Case—2: Two similar charge Q, and Q, are placed along a straight line at a distance x from each other and a third dissimilar chargeg is placed in between them as shown below A B 66—§ 2 0 & se, <—_5__ se 4 > — — x Charge q will be in equilibrium if |F,| =|F,| Siu Same short trick can be used here to find the position of charge g as we discussed in Case-l ie., x x xy= and x, = TH YGIQ * 144Q7Q It is very important to know that magnitude of charge g can be determined if one of the extreme charge (either Q, or Q, ) is in equilibrium ze. if Q, is in equilibrium then |q| =Q,(x/x? and if Q, is in equilibrium then |q|= Q,(x,/x)" It should be remember that sign of g is opposite to that of Q, rQ) Case-3: Two dissimilar charge Q, and Q, are placed along a straight line at a distance x from each other, a third charge q should be placed out side the line joining Q, and Q, for it to experience zero net force. Cet2,)<|2,) Short Trick : For it’s equilibrium. Charge g lies on the side of charge which is smallest in magnitude and x ¥Q/Q-1 @ Equilibrium of suspended chargein an electric field @ ~~ Freely suspended charged particle : To suspend a charged a particle freely in air under the influence of electric field it’s downward weight should be balanced by upward electric force for example if a positive charge is suspended freely in an electric field as shown then or or mg tet + FH ++ + mg =n 4 E Inequilibrium QE=mg > E = ‘© Ka be In the above case if direction of electric field is suddenly reversed in any figure then acceleration of charge particle at that instant will be a= 2g. Case -1: If some charge say + Q is given to bob and an electric field E is applied in the direction as shown in figure then equilibrium position of charged bob (point charge) changes from O to O*, @ mg On displacing the bob from it’s equilibrium position 0" It will oscillate under the effective acceleration g’, where mg!= me)’ +(QE)’ >= ye’ +(QE/my Hence the new time period is T, = anf g — a (e? +(QB/m)' Since g‘> g, hence T, T, =20 ie. time period of pendulum will decrease. Case—2: If electric field is applied in the downward direction then. ©) Effective acceleration g’=g + QE/m So newtime period T, = 27, g+(QE/ T,

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