2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1


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1  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation of this euation gives the wok done in displacing a unit positive chage fom P to as W E dl P This wok depends only on the initial and final positions of the unit chage and not on the path followed by it. Hence, wok done in moving a chage along a closed path is eual to zeo. Thus electic field like gavitational field is a consevative field.. Electostatic Potential The wok done by the electic field in moving a unit positive electic chage fom an abitaily selected efeence point θ, which may be inside o outside the field, to point P is given by W P P E dl θ Fo the selected efeence point, the value of W P depends only on the position of point P and not on the path followed in going fom efeence point to point P. Let θ be at infinity. The electic field at infinite distance due to finite chage distibution will be zeo. The electic field due to an infinitely long chaged plane at infinite distance will not be zeo. Howeve, in pactice, one cannot have such a chage distibution. The wok done in a diection, opposing the electic field in binging a unit positive chage fom an infinite position to any point in the electic field is called the static electic potential ( V ) at that point. Its sign is taken as negative as the wok done is in a diection opposite to the electic field. Thus, wok done in binging a unit positive chage fom infinity to points P and will be P V ( P )  E dl and V ( )  E dl P P V ( )  V ( P )  E dl + E dl E dl + E dl  E dl P This euation gives the electic potential of point with espect to point P. Its unit is volt ( joule / coulomb ) denoted by V and its dimensional fomula is M L T  3 A .
2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page.3 Electic Potential Enegy and Potential Diffeence A stationay electic chage at infinity has no enegy ( kinetic o potential ) associated with it. If a unit positive chage is bought fom infinity to an abitay point P in the electic field such that it has no velocity at that point then, the field being consevative, wok done on it is stoed with it in the fom of potential enegy and is called the electic potential of the point P and is given by V ( P )  P E dl If the electic chage is of magnitude instead of unity, then the wok done is called the potential enegy of the chage at point P and is given by U ( P ) V ( P )  P E dl The oiginal electic field o the aangement of chages in the field should emain unaffected by binging the electic chage o the unit chage fom an infinite distance to the point in the electic field. Geneally, one needs to calculate the potential diffeence o the diffeence in potential enegy of chage when it is moved fom P to which is given by U ( )  U ( P )  P E dl It should be noted that this potential enegy o the potential enegy change is associated not only with the chage but also with the entie chage distibution which gives ise to the electic field.. Electic Potential due to a Point Chage The electic field at any point A having as its positionvecto due to a point chage placed at the oigin of the coodinate system is given by E ( ) k The electic potential at point A as shown in the figue is given by A V ( A )  E dl Moving in the adial diection fom infinity to point A, dl d
3  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 3 A k V ( A ) ( d )  A d  k  k k  V ( A ) V ( ) The electic potential is positive if is positive and negative if is negative. Hence, fo negative chage negative sign should be used fo in the above euation..5 Electic Potential in the Field of an Electic Dipole Let the two chages  and + be placed at A and B espectively with a distance a between them. Let P be a point at a distance fom the cente of this dipole and making an angle θ with its diection such that AP  and BP +. The electic potential at P is given by V ( ) π If > > a, then +  and a cos θ As the atomic dipoles ae of vey small magnitude, this appoximation holds good. V ( ) As p cos θ a cosθ p, V ( ) p cosθ p ( p a ) ( fo l l > > a ) Note: ( ) If and a 0 in p a, then the dipole is called a point dipole. ( ) The above euation gives the exact value of the electic potential fo a point dipole and an appoximate value fo a dipole system othe than a point dipole. p ( 3 ) Fo any point along the axis of the dipole, θ 0 o π and V ±. ( ) Fo a point along the euato of the dipole, θ π / and V 0. ( 5 ) In the case of a dipole, the electic potential vaies as / and not as / as in the case of a point chage.
4  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page.6 Electic Potentials due to Diffeent Types of Chage Distibutions.6 ( a ) Discete Distibution of Chages: The point electic chages,, n have position vectos,,... n espectively with espect to the oigin. The total electic potential at point P having position vecto due to the entie chage distibution is eual to the algebaic sum of the electic potential due to each of the chages of the system and is given by V V + V + + V n l  l + l  l n l  n l V ( ) n i k i l  l i.6 ( b ) Continuous Distibution of Chage: Let ρ ( ' ) volume chage density fo continuous chage distibution in some egion. dτ small volume element having position vecto, ' The electic potential due to some volume element at some point P having position vecto,, is given by dv ρ ( ' ) d τ ' l  ' l Integating, we get the electic potential at the point P due to entie chage of the system as V ( ' ) π ε 0 V ρ ( ' ) dτ' l  ' l Fo constant chage distibution, ρ ( ' ) ρ can be taken as constant.
5  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 5.6 ( c ) Unifomly Chaged Spheical Shell: Fo a spheical shell of adius R caying chage on its suface, the electic potential at a point outside the shell at a distance fom its cente can be calculated by teating the entie chage of the shell as if concentated at its cente and is given by V ( R ) The electic field inside the sphee is eual to zeo. So the electic potential will be the same fo all points inside the spheical shell and its value will be eual to the electic potential on the suface of the shell. Thus, the electic potential at the suface of the spheical shell and inside will be constant and is given by V R ( R ).7 Euipotential Sufaces If the electic potential at evey point of any imaginay suface in an electic field is the same, then such a suface is called an euipotential suface. As the electic potential at a distance due to a point chage is given by V, all concentic spheical sufaces with the chage at the cente ae euipotential sufaces. The electic intensity vecto at any point is always pependicula to the euipotential suface passing though that point. This can be poved as unde. Wok done fo small displacement dl of a unit positive chage along the euipotential suface in a diection opposite to the electic field  E dl potential diffeence between the two points 0 ( as the points ae on euipotential suface ) ( E ) ( dl ) cos θ 0, whee θ is the angle between E and dl. Since E 0 and dl 0, cos θ 0 cos θ π / which shows that E and dl ae pependicula to each othe. As dl is tangential to the suface of the sphee, E is pependicula to the spheical suface which is euipotential. Fo unifom electic field, euipotential sufaces will be planes pependicula to the electic field lines.
6  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 6.8 The Relation between Electic Field and Electic Potential The electic potential diffeence between two close points P and is given by dv  E dl If E E x i and dl dx i + dy j + dz k, then dv  ( E x i ). ( dx i dy j + dz k + ) dv  E x dx and dv dx  Ex Similaly, if the electic field exists along the Y and Z diections, then dv  Ey and dy dv  Ez dz If the electic field vecto has all the thee components ( x, y, z ), then the elation between the electic potential and the electic field can be given as x  E x, y  E y and z  E z o E  ( x i + y j + z k ) Hee, x y espectively. and z ae the patial deivatives of V ( x, y, z ) with espect to x, y and z While taking the patial deivative of V ( x, y, z ) with espect to x, the emaining vaiables y and z ae taken constant. Such a deivative of V is called its patial deivative with espect to x and is denoted by. x In geneal, if the electic field exists along the diection, then  E. The electic field is a consevative field and the above euations can only be used fo a consevative field..9 Potential Enegy of a System of Chages Wok done in binging electic chages fom infinity to thei espective positions in a system of chages gets stoed in the fom of electic potential enegy of the system of chages. The potential enegy of a system of two chages and is given as dv d U k
7  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 7 If a thid chage 3 is intoduced in the system, then the potential enegy of the system of thee chages will be U U + U 3 + U 3 In geneal, if the system is made up of n electic chages, U n Uij i i < j n i i < j ki j ij.9 ( a ) Potential Enegy of an Electic Dipole in a Unifom Electic Field The potential enegy of an electic dipole AB in a unifom electic field ( E ) as shown in the figue is eual to the algebaic sum of the potential enegy of the two chages of the dipole. Taking the electic potential nea the negative chage to be zeo, the potential enegy of the dipole will be eual to the potential enegy of the positive chage. Let V be the change in electic potential as we move fom A to B. V  E x  E ( AC )  E ( a cos θ ) the potential enegy of electic chage at point B is U V  E (.a ) cos θ  E p cos θ  E p ( i ) When the electic dipole is paallel to the field,  E p  pe which means that the dipole has minimum potential enegy ( as it is moe negative ) which is also the euilibium position of the dipole. ( ii ) If θ π, the dipole has maximum potential enegy, pe, and is in an unsteady position. ( iii ) When the electic dipole is pependicula to the field ( θ π / ), the potential enegy of the dipole is zeo..0 Conductos and Electic Fields When a conducting mateial is placed in a unifom electic field as shown in the figue, fee electons migate in a diection opposite to the electic field and get deposited on one side of the metal suface while the positive chage gets deposited on the othe side of the conducto. This poduces an electic field inside the conducto and the migation of chages stops when the intenal electic field becomes eual to the extenal field.
8  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 8 If we daw a Gaussian suface inside the conducto as shown in the figue, then since the electic field on it is zeo, the net electic chage enclosed by it is also zeo. The impotant conclusions ae: ( ) Stationay electic chage distibution is induced on the suface of the conducto. ( ) Both the electic field and the net electic chage inside the conducto ae zeo. ( 3 ) At evey point on the oute suface of the conducto, the electic field is pependicula to the suface. This is so because the electic chage on the suface is stationay which means that no tangential foce acts on it, thus poving that the electic field on the suface has no tangential component. Conside anothe example of a hollow conducto placed in an extenal electic field. Hee also, the electic chages deposit on the oute sufaces and the electic field inside is zeo as thee is no chage inside. This phenomenon is called Electostatic Shielding. When a ca is stuck by lightning, the peson sitting inside is saved fom lightning as the ca is hollow and acts like an electostatic shield. Electic field inside a chaged conducto which is NOT in an electic field is also zeo. Conside a Gaussian suface close to the suface of the conducto as shown by boken line in the figue. The line integation of the electic field along the Gaussian suface being zeo, the net electic chage enclosed by it is also zeo. This shows that in a chaged conducto, the electic chage gets distibuted on the oute suface of the conducto. As the electic chages ae stationay, the diection of the electic field will be pependicula to the suface of the conducto as shown in the figue and its magnitude will be. To explain it, conside a pillbox shaped Gaussian suface on the suface of the conducto as shown in the above figue. The chage enclosed by the Gaussian suface A. The total flux passing though this suface A E by Gauss s law, A E A E If is not unifom along the suface, its pope value at the point should be used to calculate value of E at that point. If a positive electic chage is placed in the cavity of the conducto as shown in the adjoining figue, it induces chages on the inne and oute sufaces of the conducto in such a way that the field will be zeo in the inteio potion of the conducto.
9  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 9. Capacitos and Capacitance When positive electic chage on an isolated conducting sphee shown in figue is gadually inceased, electic potential on its suface and electic field in its vicinity incease. When the electic field becomes stong, it ionizes the suounding ai which causes chage on the sphee to leak and the chage on the sphee cannot be inceased futhe. Duing this pocess, the atio of chage ( ) and the electic potential ( V ) of the sphee emains constant. This atio, / V, is called its capacitance ( C ). Fig. Fig. Fig. 3 To incease capacitance C of the sphee, anothe isolated conducting sphee is bought nea it on which chage is induced as shown in figue. On eathing, the positive chage gets neutalized as shown in figue 3. The negative chage induced on the second sphee educes the electic potential of the fist sphee theeby inceasing its chage stoage capacity. The atio / V of the chage on the fist sphee and the potential diffeence between the two sphees is still constant and is called the capacitance C of the system. The value of C depends on the dimensions of the sphees, the distance between the two sphees and the medium between them. The aangement in which two good conductos of abitay shape and volume, ae aanged close to one anothe, but sepaated fom each othe, is called a capacito. The conductos ae known as plates of the capacito. Positively chaged conducto is called the positive plate and the negatively chaged conducto the negative plate. Both the plates ae eually chaged. The chage on the positive plate is called the chage ( ) of the capacito and taking the potential diffeence between the two plates as V, capacitance of the capacito is C / V. The S.Ι. unit of capacitance is coulomb / volt which is also called faad ( F ) named afte the geat scientist Michael Faaday. The smalle units of faad ae micofaad ( µ F 06 F ) and picofaad ( pf 0  F ).. Paallel Plate Capacito This type of capacito is made by two metallic plates having identical aea and kept paallel to each othe. The distance ( d ) between the two plates is kept less as compaed to the dimensions of the plates to minimize nonunifom electic field due to the iegula distibution of chages nea the edges. Let electic chage on the capacito / A suface chage density As d is vey small, the plates can be consideed as infinitely chaged planes and the electic field between the plates can theefoe be consideed unifom.
10  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 0 The electic fields, E and E, between the plates due to positively chaged and negatively chaged plates espectively ae eual in magnitude and diection. The diection of both E and E is fom the positively chaged plate to the negatively chaged plate. The esultant electic field between the plates is, theefoe, E E + E ε + 0 A ( ) A Outside the plates, E and E being oppositely diected cancel each othe esulting in zeo electic field in this egion. The potential diffeence between the two plates is V E d A d the capacitance of the capacito, C ε 0 V da.3 ( a ) Seies Connection of Capacitos The end to end connection of capacitos as shown In the figue is called the seies connection of capacitos. Eual chage deposits on each capacito, but the p.d. between thei plates is diffeent depending on the value of its capacitance. V V + V +. + V n C C C n V When all capacitos connected in seies ae eplaced by a C C Cn single capacito of capacitance C such that the chage deposited on it is with the same voltage supply, then such a capacito is called thei euivalent capacito. V C C C + C C n The value of C is smalle than the smallest of C, C,. C n..3 ( b ) Paallel Connection of Capacitos The connection of capacitos in which positive plates of all capacitos ae connected to a single point and negative plates to anothe single point in a cicuit is called paallel connection of capacitos as shown in the figue. In such a connection, chage accumulated on each of the capacitos is diffeent depending on the value of its capacitance, but the p.d. acoss all is the same.
11  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page Thus, total chage ( C + C + C 3 +. ) V When all capacitos connected in paallel ae eplaced by a single capacito of capacitance C such that the chage deposited on it is with the same voltage supply, then such a capacito is called thei euivalent capacito. Its value is C / V C + C + C Enegy Stoed in a Chaged Capacito The wok done in chaging the capacito gets stoed in it in the fom of potential enegy. The electic field on one plate due to chage and chage density on it The potential enegy of one plate at a distance d fom the othe plate U E ( electic p.d. between them ) ( electic chage on the second plate ) d ( taking efeence potential on the fist plate as zeo ). Putting U E, A d A A / d C Putting CV o C / V, we get U E C CV V Enegy Stoed in a Capacito in tems of Enegy Density of Electic Field If A aea of the plates of the capacito d sepaation between the plates then enegy density of electic field in the capacito ρ E enegy stoed pe unit volume U E CV ε 0 A V V ε A d A d d 0 A d d V ε E ( E ) 0 d The above euation gives enegy stoed in a capacito in tems of the enegy stoed in the electic field between the two plates. This is a geneal esult which holds tue fo an electic field due to any chage distibution..5 Dielectic Substances and thei Polaization A nonconducting mateial is called a dielectic.
12  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page The dielectic does not possess fee electons like a conducting mateial. Intoduction of a dielectic between the two paallel plates of a capacito consideably inceases the capacitance of a capacito. Dielectic mateials ae of two types: ( ) Pola and ( ) Nonpola. The atoms of a dielectic ( like HCl, H O ) that have pemanent dipole moment is called a pola dielectic. The atoms of a dielectic ( like H, O ) that do not have pemanent dipole moment is called a nonpola dielectic..5 ( a ) Nonpola Atoms ( o Molecules ) placed in a Unifom Electic Field: The centes of the positive and the negative chages coincide in a nonpola dielectic atom o molecule. So it does not have pemanent dipole moment. If such an atom o molecule is placed in a unifom electic field ( E ), the centes of the positive and negative electic chages get sepaated due to electic foce acting on them in opposite diections. This is known as polaization of the atom. If the electic field is not vey stong, then the dipole moment, p, of the induced dipole is diectly popotional to E. p α E, whee constant α is called the atomic ( o molecula ) polaizability..5 ( b ) Dielectic in a Capacito A dielectic slab of pola mateial is placed between the paallel plates of a capacito as shown in the figue. The toue due to the electic field acting on atomic dipoles aligns them in the diection of the electic field. In a solid dielectic mateial, its atoms pefom oscillatoy motion, wheeas in liuid o gas dielectic mateials, they pefom linea motion. The dipoles ae in euilibium and completely aligned as shown in the figue when its electic potential enegy,  p E, euals the themal 3 enegy, k T, due to absolute tempeatue T. It can be seen in the figue that the atomic dipoles line up such that the opposite polaity face each othe canceling each othe s effect. The net chage of the system is the chage of the capacito plates. The dipole close to the positive plate has negative chage lined up facing it and vice vesa. These induced electical chages ae called bound chages. The electic field between the capacito plates is E f E ε f 0  f b ( ) in the absence of the dielectic medium and is ( ) in the pesence of the dielectic slab,
13  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 3 whee f is the fee ( contollable ) suface chage density on the suface of the capacito plates and b is the bound suface chage density on the suface of the dielectic slab. b is taken negative as the suface chages of the dielectic slab have opposite polaity as compaed to the chages of the capacito plates. Dete mining b The dipole moment of the dielectic slab having suface aea A and length L is P t b A L P t A L P t b P, V whee P t / V is the dipole moment of the dielectic slab pe unit volume and P is called the intensity of polaization. If the electic field is not vey stong, the atio of P and the electic field is a constant. P χ e E This atio χ e is called the electic susceptibility of the dielectic mateial and its value depends on the dielectic mateial and its tempeatue. P b χe E  0 E ε f P 0 ε E ( + χ e ) E Hee, ε 0 χ ε ε f  0 ε E f f 0 0 Ef + χ e χ e E [ using euations ( ) and ( ) ] + e ε is called the pemittivity of the medium and its value depends on the type of the dielectic mateial and its tempeatue. Its unit is C N  m . E ε Ef ε E ε f is the elative pemittivity Eε f E f K ε and is also called dielectic constant K. This shows that when a dielectic mateial having dielectic constant K is placed as a medium between the capacito plates, the electic field is educed by a facto of K and the capacitance of the capacito becomes C ' ε A d K A d K C Thus when a dielectic medium having dielectic constant K is intoduced in a capacito, the value of its capacitance inceases to K times the oiginal capacitance.
14  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page Fom euation ( ), we have E + P f ( b P ) Hee, E and P ae electic and polaization fields espectively which ae both vecto uantities. we can wite ε0 E + P D, whee D is called the displacement field..6 VandeGaf Geneato The pinciple of this machine is as unde. A conducting sphee having chage and adius is kept centally inside a conducting shell having chage and adius R ( < R ). The electic potential on the spheical shell of adius R and on the suface of the sphee of adius ae espectively V R k k + and V R R k + R k Hence, the potential diffeence between the sufaces of the two sphees is V  V R k k +  R k k  k  R R R The above euation shows that the smalle sphee is at a highe potential as compaed to the lage spheical shell. So if the two ae connected though a conducto, then the electic chage will flow fom the smalle sphee to the lage spheical shell. So if chage is tansfeed continuously to the smalle sphee in some way, then it will accumulate on the lage shell theeby inceasing its electic potential to a vey lage value. VandeGaf used this pinciple to constuct a geneato known as VandeGaf geneato to geneate seveal million volts of electic potential. As shown in the figue, a nonconducting belt is diven acoss two pulleys. The lowe pulley is connected to a moto and the uppe one is suounded by a spheical shell. Positive electic chage is tansfeed to the belt nea the lowe pulley using a dischage tube and a bush with shap edges. The electic chage moves to the uppe pulley and is deposited on the oute shell with a metallic bush. Thus electic potential of the ode of 6 to 8 million volts is geneated on the oute shell. The highly intense electic field poduced in this device can acceleate electic chages which can be used to study the composition of matte at the micoscopic level.
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