ELECTRIC POTENTIAL AND CAPACITANCE

Transcription

1 ELECTRIC POTENTIAL AND CAPACITANCE PEPC. Intoduction : Q. What is the natue of electostatic foce o Coulomb s foce? Solution : Coulomb s foce between two stationay chages is a consevative foce. Q. Does the wok done by an electostatic field depend on the path? Explain. Solution : As the electostatic field is consevative field, i.e., the wok done by electostatic field is path independent. Q. What is the wok done by electostatic foce fo closed path? Solution : Zeo Q. Can absolute value of potential enegy be defined o significant? Explain. Solution : No, thee is only the significance of diffeence of potential enegy because it equals to the wok done by the extenal foce in binging the chage fom one point R to the point P in the pesence of field due P to any chage configuation i.e., U = U P U R = W RP = F. d. Q. Define electostatic potential enegy diffeence. Solution : It equals to the wok done by the extenal foce in binging the chage q fom one point R to the P point P in the pesence of field due to any chage configuation i.e., U = U P U R = W RP = F ext. d. R Q. Define electostatic potential enegy. Wite down its units and dimensional fomula. Solution : Electostatic potential enegy of chage q at a point P (in the pesence of field due to any chage configuation) is the wok done by the extenal foce (equal and opposite to the electic foce) in binging (infinitesimally slow constant speed) the chage q fom infinity to that point. Mathematically, W P = U P U = U P (The potential enegy at infinity is taken as zeo). The unit of electostatic potential enegy is joule and dimensional fomula [ML T ] Q. Conside a chage Q placed (fixed) at the oigin. A test chage q is moved vey slowly by an extenal agent fom infinity to the point P at distance fom Q. Deive the expession fo wok done by extenal agent in this pocess? Deive the expession fo electostatic potential enegy of chage q? Solution : Conside a point chage Q at the oigin. R ext At some intemediate point P on the path, the electostatic foce on chage q is a unit vecto along OP. Wok done against this foce fom to d is Q q 4 ˆ, whee ˆ is Q q dw 4 d The negative sign appeas because fo d, dw is positive. Total wok done (W) by the extenal foce is obtained by integating fom (magnitude ) to (magnitude ). Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

2 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857 Qq Qq Qq W d This wok done W is the expession fo electostatic potential enegy of chage q.. Electostatic Potential : Q. Define electostatic potential diffeence. Find an expession fo it due to a point chage? PEPC Solution : It equals to the wok done by the extenal foce in binging the unit chage fom one point R to the point P in the pesence of field due to any chage configuation. Mathematically, V whee V P and V R ae the electostatic potentials at P and R, espectively. P UP UR VR, q Conside a point chage Q at the oigin. To detemine the potential diffeence between the points R (with position vecto fom the oigin) and P (with position vecto fom the oigin) due to this chage, we must calculate the wok done in binging a unit positive test chage fom R to P. At some intemediate point P on the path, the electostatic foce on a unit positive chage is whee ˆ is a unit vecto along OP. Wok done against this foce fom to d is Q 4 ˆ, Q dw 4 d The negative sign appeas because fo d, dw is positive. Total wok done (W) by the extenal foce is obtained by integating fom (magnitude ) to (magnitude ). Q Q Q W d This wok done W equals to V P V R Q. Define electostatic potential. Is it a vecto o scala quantity? Wite down its unit and dimensional fomula. Solution : The electostatic potential (V) at any point due to a chage distibution in a egion with electostatic field is the wok done in binging a unit positive chage (without acceleation) fom infinity to that point. Mathematically, V potentials at P and, espectively. P V U P U q U q P, whee V P and V (= ) ae the electostatic It is a scala quantity. The unit of electostatic potential is joule/coulomb o volt. The dimensional fomula is [ML T 3 A ] Q. Can absolute value of potential be defined o significant? Explain. Solution : No, thee is only the significance of diffeence of potential because it equals to the wok done by the extenal foce in binging the unit chage fom one point R to the point P in the pesence of field due to any chage configuation.

3 Q. While defining potential why is infinity taken as a efeence point? PEPC 3 Solution : Infinity is taken as a efeence point because at infinity potential due to any chage configuation equals to zeo..3 Potential Due to a Point Chage : Q. Deive the expession fo electostatic potential due to a point chage Q at the distance. Solution : Conside a point chage Q at the oigin. To detemine the potential at any point P (with position vecto fom the oigin) due to this chage, we must calculate the wok done in binging a unit positive test chage fom infinity to the point P. At some intemediate point P on the path, the electostatic foce on a unit positive chage is whee ˆ is a unit vecto along OP. Wok done against this foce fom to d is Q 4 ˆ, Q dw 4 d The negative sign appeas because fo d, dw is positive. Total wok done (W) by the extenal foce is obtained by integating fom to. W Q Q Q d This, by definition is the potential at P due to the chage Q Q V(). 4 Q. Why the electostatic potential due to a single chage is spheically symmetic? Solution : As the electostatic potential due to a single chage is invesely popotional to and at equal distance fom the chage, value of v emains same. Q. Daw the gaph fo the vaiation of electostatic potential and electostatic field due to a point chage with distance. Solution : Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

4 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857 PEPC 4 Q. A point chage q is placed at O. Is V P V Q, positive o negative when (i) q >, (ii) q <? Justify you answe. Solution : V P V Q equals to the wok done by an extenal agent to move a unit positive chage fom Q to P in the pesence of othe chages. (i) when q >, the wok done by an extenal agent to move a unit positive chage fom Q to P is positive hence V P V Q is positive. (ii) when q <, the wok done by an extenal agent to move a unit positive chage fom Q to P is negative hence V P V Q is negative. Q. If the electostatic potential at any point due to cetain chage configuation equals to V then find the wok done by the extenal agent in binging a chage q fom infinity to this point without acceleation (vey slowly)? Solution : The electostatic potential (V) at any point in a egion with electostatic field is the wok done in binging a unit positive chage (without acceleation) fom infinity to that point. Hence the wok done by the extenal agent in binging a chage q fom infinity to this point without acceleation (vey slowly) equals to qv. Q. Define one volt. Solution : Electostatic potential at a point is said to be one volt, when one joule of wok is done in moving one coulomb of positive chage fom infinity to that point by an extenal agent (against the electostatic foce of the field) without acceleation. Q. (a) Calculate the potential at a point P due to a chage of 4 7 C located 9 cm away. (b) Hence obtain the wok done in binging a chage of 9 C fom infinity to the point P. Does the answe depend on the path along which the chage is bought? [NCERT solved example.] Solution : (a) 4 4 V (b) 8 5 J, wok done will be path independent..4 Potential Due to an Electic Dipole : Q. Deive an expession fo the electic potential at any point along (i) axial line (ii) equatoial line of an electic dipole. Solution : Electic potential due to dipole along axial line The potential due to +q at P is The potential due to q at P is q. 4 ( a) q 4 ( a) Fom pinciple of supeposition, the net potential at P is q q q(a) p V 4 ( a) 4 ( a) 4 ( a ) 4 ( a Hee p is the dipole moment of dipole equals to q(a). ) Fo shot dipole, >> a V p 4

5 PEPC 5 Potential at equatoial line The electostatic potential at point P due to +q is 4 q a Hence, fom pinciple of supeposition, the net potential at P equals to zeo. and due to q is 4 q a Q. Deive an expession fo the electic potential at any point due to electic dipole. Discuss this esult fo axial line and equatoial line. Solution :. Let us take the oigin at the cente of the dipole. We have to calculate electic potential at any point P whee OP. Let the distance of P fom chage q be and distance of P fom chage +q be. Electostatic Potential at P due to q chage is q V 4 and Electostatic potential at P due to +q chage is Theefoe, potential at P due to the dipole, V q 4 V = V + V Now, by geomety q V...() 4 = + a a cos, and = + a + a cos Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

6 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857 We may ewite a PEPC 6 a cos a a If a < <, is small, can be neglected. a cos a cos / a cos / Using Binomial appoximation, ( + x) n = + nx fo x << a cos Similaly, Putting these values in () a cos q V 4 q V 4 q a cos V 4 pcos V 4 a cos a cos a cos a cos As p cos = p. ˆ, whee ˆ is unit vecto along the position vecto OP, p. ˆ Electostatic potential at P due to a shot dipole (a < < ) is V 4 On the dipole axis, = o V 4 p Positive sign fo = and negative sign fo =. At any point in the equatoial plane, ;cos cos. V = i.e., electostatic potential at any point in the equatoial plane of dipole is zeo.

7 PEPC 7 Q. What is the diffeence between the electic potential of a dipole fom that due a single chage? Solution : (i) The potential due to a dipole depends not just on but also on the angle between the position vecto and the dipole moment vecto p. (ii) The electic dipole potential falls off, at lage distance, as /, not as /, chaacteistic of the potential due to a single chage. Q. What do you mean by axially symmetic about dipole moment of potential due to an electic dipole? Solution : If you otate the position vecto about p, keeping (between and p) fixed, the points coesponding to P on the cone so geneated will have the same potential as at P. Q. What do you mean by cylindical symmety of electical potential due to an electic dipole? Solution : It means potential due to electic dipole will be same at evey point on the suface of a ight cicula cylinde imaginated with the electic dipole as the axis..5 Potential due to a system of Chages : Q. Which pinciple is used to find the potential due to system of chages and deive the potential due to system of chages? o Conside a system of chages q, q, q 3... q n with position vectos,... n. Find the expession fo the potential at point P of position vecto P. Solution : Supeposition pinciple is used to find the potential due to system of chages. Conside a system of chages q, q, q 3... q n with position vectos,... elated to some oigin O. n The potential V at P due to the chage q is q (P P ), whee 4 P is the distance between q and P. V P Similaly, the potential V at P due to q and V 3 due to q 3 ae given by q (P P ) q3 4, V3 (3P P 3 ) P 4 V whee P and 3P ae the distances of P fom chages q and q 3, espectively; and so on fo the potential due to othe chages. By the supeposition pinciple, the potential V at P due to the total chage configuation is the algebaic sum of the potentials due to the individual chages V = V + V V n 4 q q q... n P P np Q. What is the electostatic potential due to unifomly chaged spheical shell? Solution : Fo a unifomly chaged spheical shell, the electic field outside the shell is as if the point chage is concentated at the cente. Thus, the potential outside the shell is given by 3P V 4 q ( R) Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

8 PEPC 8 whee q is the total chage on the shell and R its adius. The electic field inside the shell is zeo. This implies that potential is constant inside the shell (as no wok is done in moving a chage inside the shell), and, theefoe, equals its value at the suface, which is V 4 q ( R R) Q. Daw the gaph fo the vaiation of potential due to unifomly chaged spheical shell with the distance fom the cente. Solution : Q. Two chages 3 8 C and 8 C ae located 5 cm apat. At what point on the line joining the two chages is the electic potential zeo? Take the potential at infinity to be zeo. [NCERT solved example.] Solution : Electic potential is zeo at 9 cm and 45 cm away fom the positive chage on the side of the negative chage. Q. Figues (a) and (b) show the field lines of a positive and negative point chage espectively. (a) Give the signs of the potential diffeence V P V Q ; V B V A. (b) Give the sign of the potential enegy diffeence of a small negative chage between the points Q and P; A and B. (c) Give the sign of the wok done by the field in moving a small positive chage fom Q to P. (d) Give the sign of the wok done by the extenal agency in moving a small negative chage fom B to A. (e) Does the kinetic enegy of a small negative chage incease o decease in going fom B to A? [NCERT solved example.3] Solution : (a) +, + (b) +, + (c) (d) + (e) decease..6 Equipotential Suface : Q. Define equipotential suface. Solution : An equipotential suface is a suface with a constant value of potential at all points on the suface. Q. Daw equipotential suface fo a point chage. What is the shape of equipotential suface fo point chage? Solution : The shape of equipotential suface fo a point chage is spheical. Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

9 PEPC 9 Q. Why equipotential suface though a point is nomal to the electic field at that point fo any chage configuation? Solution : If the field wee not nomal to the equipotential suface, it would have non-zeo component along the suface. To move a unit test chage against the diection of the component of the field, wok would have to be done. But this is in contadiction to the definition of an equipotential suface : thee is no potential diffeence between any two points on the suface and no wok is equied to move a test chage on the suface. The electic field must, theefoe, be nomal to the equipotential suface at evey point. Q. Can two equipotential suface eve intesect each othe? Comment Solution : No, they cannot intesect at any point. If they will intesect then at the point of intesection the potential fo each suface will be the same which is not possible. Q. If an electic field is along the x-axis then what will be the shape of equipotential suface? Solution : If an electic field is along the x-axis then the shape of equipotential suface is planes paallel to the yz plane. Q. What will be the wok done by an extenal agent to move a chage q fom one point of an equipotential suface of potential V to anothe point of anothe equipotential suface of potential V? Solution : q (V V ) Q. What will be the wok done by an extenal agent to move a chage fom one point to anothe point on the same equipotential suface? Give eason. Solution : The wok done by an extenal agent to move a chage fom one point to anothe point on the same equipotential suface equals to zeo because on an equipotential suface, the potential at each point emain same. Q. State main chaacteistics of an equipotential suface. Solution : The main chaacteistics of an equipotential suface ae as given below : (i) An equipotential suface is a suface with a constant value of electic potential at all points on the suface. (ii) No wok is equied to be done to move a chage on an equipotential suface. (iii) Fo any chage configuation, equipotential suface though a point is nomal to the electic field at that point. (iv) No two equipotential sufaces can eve intesect each othe. (v) Relative closeness of equipotential sufaces having a given potential diffeence means a egion of stonge electic field. Q. Daw the equipotential suface fo (a) unifom electic field (b) dipole (c) two identical positive chages. Solution :(a) (b) Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

10 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857 PEPC (c) Q. What is the elation between electic field and electic potential? Deive it. Solution : Conside two closely spaced equipotential sufaces A and B with potential values V and V + V, whee V is the change in V in the diection of the electic field E. Let P be a point on the suface B. l is the pependicula distance of the suface A fom P. Imagine that a unit positive chage is moved along this pependicula fom the suface B to suface A against the electic field. The wok done in this pocess is E l. This wok equals the potential diffeence V A V B. Thus, E l = V (V + V) = V V i.e., E l V V Since V is negative, V = V, we can ewite as E l l Q. Wite down two conclusion concening the elation between E and potential OR what is the physical meaning of potential gadient? Is it a scala o vecto quantity. Solution : We aive at two impotant conclusions concening the elation between electic field and potential : (i) Electic field is in the diection in which the potential deceases steepest. (ii) Its magnitude is given by the change in the magnitude of potential pe unit displacement nomal to the equipotential suface at the point. Potential gadient is a vecto quantity. Q. In a paticula egion in space the electic potential has a constant high value. What is the value of electic field thee? Solution : As potential is constant, hence, value of electic field stength is zeo in that egion in space in that diection.

11 PEPC Q. Is electostatic potential necessaily zeo at a point whee electic field stength is zeo? Illustate you answe. Solution : No, if the electic field in the space is zeo then the potential in that space may have some non-zeo constant value. Fo e.g., inside the coducto electic field is zeo but the potential will exists. Q. At which of the points A, B and C in a unifom electostatic field as shown, will the electic potential be (a) minimum (b) maximum Solution : V A = V B > V C Q. In a space the electic potential is changing with distance accoding to V = k. Find the electic field. Solution : k Q. The electostatic potential V is changing with distance accoding to the following gaph. Daw the gaph fo the vaiation of electic field with distance. Solution : Q. The electic field in a space due to cetain chaged distibution is invesely popotional to distance. How the electostatic potential depends on distance? Solution : popotional to ln Q. A metal sphee with a chage Q is suounded by an unchaged concentic thin spheical shell. The potential diffeence between them is V. If the shell is now given an additional chage Q, what is the new potential diffeence between them? Solution : V Q. We know that electic field is discontinuous acoss the suface o a chaged conducto. Is electic potential also discontinuous thee? Solution : No.7 Potential Enegy of a System of Chages : Q. Deive the expession fo potential enegy of a system of thee point chages q, q and q 3 placed at the position, and 3 elative to some oigin O? Solution : Conside a system of thee point chages q, q and q 3 placed at the position, and 3 elative to some oigin O. A system of thee chages q, q and q 3 located at,, 3 espectively. Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

12 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857 PEPC To bing q fist fom infinity to, no wok is equied. Next we bing q fom infinity to. As befoe, wok done in this step is 4 qq. Wok done next in binging q 3 fom infinity to the point 3 is 4 q q 3 q q The total wok done in assembling the chages at the given locations is obtained by adding the wok done in diffeent steps, U 4 q q qq q q Q. The potential enegy is chaacteistic of the pesent state of configuation, and not the way the state is achieved. Comment. Solution : The consevative natue of electostatic foce (o equivalently, the path independence of wok done), the expession fo potential enegy, is independent of the manne in which the configuation is assembled i.e., the potential enegy is chaacteistic of the pesent state of configuation, and not the way the state is achieved. Q. Fou chages ae aanged at the cones of a squae ABCD of side d, as shown (a) Find the wok equied to put togethe this aangement. (b) A chage q is bought to the cente E of the squae, the fou chages being held fixed at its cones. How much exta wok is needed to do this? [NCERT solved example.4] q Solution : (a) 4 4 d (b) Zeo.8 Potential Enegy in an Extenal Field : Q. What is the potential enegy of a chage q in a given field? Solution : The wok done in binging the chage q fom infinity to the point P, at which the potential is V due to the field, is qv and this wok is stoed in the fom of potential enegy. If the point P has position vecto elative to some oigin, we can wite : Potential enegy of q at in an extenal field = qv ( ) whee V( ) is the extenal potential at the point.

13 Q. Define one electon volt. PEPC 3 Solution : If an electon with chage q = e = =.6 9 C is acceleated by a potential diffeence of V = volt, it would gain enegy of qv =.6 9 J. This unit of enegy is defined as electon volt o ev, i.e., ev =.6 9 J. Q. What is the value of kev, MeV, GeV & TeV in joule? Solution : kev = 3 ev =.6 6 J, MeV = 6 ev =.6 3 J, GeV = 9 ev =.6 J and TeV = ev =.6 7 J. Q. What is the potential enegy of a system of two chages q and q located at and, espectively, in an extenal field? Solution : Fist, we calculate the wok done in binging the chage q fom infinity to. Wok done in this step is q V( ). Next, we conside the wok done in binging q to. In this step, wok is done not only against the extenal field E but also against the field due to q. Wok done on q against the extenal field = q V( ) Wok done on q against the field due to q = qq 4 whee is the distance between q and q. By the supeposition pinciple fo fields, we add up the wok done on q against the two fields ( E and that due to q ) = q V( Thus, Potential enegy of the system qq ) 4 = the total wok done in assembling the configuation = q V( ) q V( qq ) 4 Q. (a) Detemine the electostatic potential enegy of a system consisting of two chages 7 µc and µc (and with no extenal field) placed at ( 9 cm,, ) and (9 cm,, ) espectively. (b) How much wok is equied to sepaate the two chages infinitely away fom each othe? (c) Suppose that the same system of chages is now placed in an extenal electic field E = A(/ ); A = 9 5 C m. What would the electostatic enegy of the configuation be? [NCERT Solved Example.5] Solution : (a).7 J (b).7 J (c) 49.3 J Q. Deive the expession fo potential enegy of a dipole in an extenal electic field E. OR Deive an expession fo the total wok done in otating the dipole though an angle in unifom electic field E. Solution : Conside a dipole chages q = +q and q = q placed in a unifom electic field E, as shown. Suppose an extenal toque ext is applied in such a manne that it just neutalises the toque p E and otates it in the plane of pape fom angle to angle at an infinitesimal angula speed and without angula acceleation. The amount of wok done by the extenal toque will be given by Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

14 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857 W ext ( )d pesin d = pe(cos cos ) PEPC 4 This wok is stoed as the potential enegy of the system. We can then associate potential enegy U() with an inclination of the dipole. Simila to othe potential enegies, thee is a feedom in choosing the angle whee the potential enegy U is taken to be zeo. A natual choice is to take = /. We can then wite, U( ) pe cos cos pecos p.e Q. A molecule of a substance has a pemanent electic dipole moment of magnitude 9 C m. A mole of this substance is polaised (at low tempeatue) by applying a stong electostatic field of magnitude 6 V m. The diection of the field is suddenly changed by an angle of 6. Estimate the heat eleased by the substance in aligning its dipoles along the new diection of the field. Fo simplicity, assume % polaisation of the sample. Solution : 3 J, So, thee is loss in potential enegy. This must be the enegy eleased by the substance in the fom of heat in aligning its dipoles..9 Electostatics of Conductos : Q. What is the electostatic field inside a conducto? Solution : Electostatic field is zeo inside a conducto at electostatic equilibium. Q. What is the diection of electostatic field at the suface of a conducto at electostatic equilibium o static situation? Give easons. Solution : Electostatic field at the suface of a chaged conducto must be nomal to the suface at evey point. If E wee not nomal to the suface, it would have some non-zeo components along the suface. Fee chages on the suface of the conducto would then expeience foce and move. In the static situation, theefoe, E should have no tangential component. Q. Thee is no net chage at any point inside the conducto, and any excess chage must eside at the suface. Explain. Solution : A neutal conducto has equal amounts of positive and negative chages in evey small volume o suface element. When the conducto is chaged, the excess chage can eside only on the suface in the static situation. This follows fom the Gauss s law. Conside any abitay volume element v inside a conducto. On the closed suface S bounding the volume element v, electostatic field is zeo. Thus the total electic flux though S is zeo. Hence, by Gauss s law, thee is no net chage enclosed by S. But the suface S can be made as small as you like, i.e., the volume v can be made vanishingly small. This means thee is no net chage at any point inside the conducto, and any excess chage must eside at the suface. Q. Why the electostatic potential is constant thoughout the volume of the conducto and has the same value (as inside) on its suface? Solution : Since E = inside the conducto and has no tangential component on the suface, no wok is done in moving a small test chage within the conducto and on its suface. That is, thee is no potential diffeence between any two points inside o on the suface of the conducto. Q. Deive the expession fo the electic field at the suface of a chage conducto of suface chage density. Solution :

15 PEPC 5 Choose a pill box (a shot pill box (a shot cylinde) as the Gaussian suface about any point P on the suface. The pill box is patly inside and patly outside the suface of the conducto. It has a small aea of coss section S and negligible height. Just inside the suface, the electostatic field is zeo; just outside, the field is nomal to the suface with magnitude E. Thus, the contibution to the total flux though the pill box comes only fom the outside (cicula) coss-section of the pill box. This equals ± ES (positive fo >, negative fo < ), since ove the small aea S, E may be consideed constant and E and S ae paallel o antipaallel. The chage enclosed by the pill box is S. By Gauss s law S ES E Including the fact that electic field is nomal to the suface, we get the vecto elation, equation, E which is tue fo both signs of. Fo >, electic field is nomal to the suface outwad; fo <, electic field is nomal to the suface inwad. Q. Conside a conducto with a cavity of any type of size and shape, with no chages inside the cavity. What is the electic field inside the cavity? Solution : The electic field inside the cavity is zeo. Q. Conside a conducto with a cavity of any type of size and shape, with no chages inside the cavity. Does the electic potential emain constant inside the cavity and why? Solution : The electic potential inside the cavity emains constant because inside the cavity electic field equals to zeo. Q. What is electostatic shielding? What is the use of this? Solution : Whateve be the chage and field configuation outside, any cavity in a conducto emains shielded fom outside electic influence; the field inside the cavity is always zeo. This is known as electostatic shielding. The effect can be made use of in potecting sensitive instuments fom outside electical influence. Q.(a) A comb uns though one s dy hai attacts small bits of pape. Why? What happens if the hai is wet o it it is a ainy day? (Remembe, a pape does not conduct electicity) (b) Odinay ubbe is an insulato. But special ubbe tyes of aicaft ae made slightly conducting. Why is this necessay? (c) Vehicles caying inflammable mateials usually have metallic opes touching the gound duing motion. Why? (d) A bid peches on a bae high powe line, and nothing happens to the bid. A man standing on the gound touches the same line and gets a fatal shock. Why? [NCERT Solved Example.7] Solution : (a) This is because the comb gets chaged by fiction. The molecules in the pape gets polaised by the chaged comb, esulting in a net foce of attaction. If the hai is wet, o if it is ainy day, fiction between hai and the comb educes. The comb does not get chaged and thus it will not attact small bits of pape. (b) To enable them to conduct chage (poduced by fiction) to the gound; as too much of static electicity accumulated may esult in spak and esult in fie. (c) Reason simila to (b). (d) Cuent passes only when thee is diffeence in potential. nˆ Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

16 . Dielectics And Polaisation : Q. What is the diffeence between dielectic and conducto? PEPC 6 Solution : Conducto contain mobile chage caies. In metallic conductos, these chage caies ae electons. In a metal, the oute (valence) electons pat away fom thei atoms and ae fee to move. These electons ae fee within the metal but not fee to leave the metal. Dielectics ae non-conducting substances. In contast to conductos, they have no (o negligible numbe of) chage caies. Q. What is the diffeence in behaviou of a conducto and a dielectic in an extenal field? Solution : The fee chage caies (electons) in conducto dift against the diection of the extenal electic field. The chage distibution in the conducto adjusts itself in such a way that the electic field due to induced chages opposes the extenal field within the conducto. This happens until, in the static situation, the two fields cancel each othe and the net electostatic field in the conducto is zeo. In a dielectic, this fee movement of chages is not possible. It tuns out that the extenal field induces dipole moment by stetching o e-oienting molecules of the dielectic. The collective effect of all the molecula dipole moments is net chages on the suface of the dielectic which poduce a field that opposes the extenal field. Unlike in a conducto, howeve, the opposing field so induced does not exactly cancel the extenal field. It only educes it. The extent of the effect depends on the natue of the dielectic. Q. What is the diffeence between pola and non-pola molecules? Solution : pola molecules (e.g., HCl, H O) have a pemanent dipole moment wheeas non-pola molecules (e.g., O, H ) have no pemanent dipole moment. Q. Is thee a net dipole moment of the dielectic consists of non-pola molecules in the pesence of extenal electic field? Explain. Solution : In an extenal electic field, the positive and negative chages of a non-pola molecule ae displaced in opposite diections. The displacement stops when the extenal foce on the constituents chages of the molecule is balanced by the estoing foce (due to intenal fields in the molecule). The non-pola molecules thus develops an induced dipole moment.the dielectic is said to be polaised by the extenal field. The induced dipole moments of diffeent molecules add up giving a net dipole moment of the dielectic in the pesence of the extenal field. Q. What is linea isotopic dielectics? Solution : The induced dipole moment is in the diection of the field and is popotional to the field stength. Substances fo which this assumption is tue ae called linea isotopie dielectics. Q. Why the net dipole moment of a substance of pola molecules equal to zeo in the absence of electic field? Solution : In the absence of any extenal field, the diffeent pemanent dipoles ae oiented andomly due to themal agiation; so the total dipole moment is zeo. Q. What is polaisation? How does it depend on the extenal electic field fo linea isotopic dielectics? Solution : A pola o non-pola dielectic developes a net dipole moment in the pesence of extenal electic field. The dipole moment pe unit volume is called polaisation denoted by P. Fo linea isotopic Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

17 dielectics, PEPC 7 P E whee e is a constant chaacteistic of the dielectic and is known as the electic e susceptibility of the dielectic medium. Q. How does the polaised dielectic modify the oiginal extenal field inside it? Solution : If a dielectic is placed in an extenal electic field, the electic field induces a net dipole moment in the dielectic. Although the net chage in the dielectic is zeo. Howeve, at the sufaces of the dielectic nomal to the electic field, thee is evidently a net chage density. As seen in figue, the positive ends of the dipoles emain unneutalised at the ight suface and the negative ends at the left suface. The unbalanced chages ae the induced chages due to the extenal field. Thus the polaised dielectic is equivalent to two chaged sufaces with induced suface chage densities, say p and p. Clealy, the field poduced by these suface chages opposes the extenal field. The total field in the dielectic is, theeby, educed fom the case when no dielectic is pesent. The electic field in the dielectic is E/K whee K is the dielectic constant of the dielectic.. Capacitos and Capacitance : Q. What is Capacitos? Solution : A capacitos is a system of two conductos sepaated by an insulato. In pactice, the two conductos have chages Q and Q with potential diffeence V = V V between them. Q is called the chage on the capacito. Q. What is the total chage on the capacito? Solution : Zeo Q. Why the potential diffeence V between the conductos of the capacitos is popotional to chage Q on the capacito? Solution : The electic field in the egion between the conductos is popotional to the chage Q. Now, potential diffeence V is the wok done pe unit positive chage in taking a small test chage fom the conducto to against the field. Consequently, V is also popotional to Q. Q. Define capacitance? Solution : The potential diffeence V between the conductos of the capacitos is popotional to chage Q on the capacito, and the atio Q/V is a constant : capacito. Q. Does the capacitance of a capacito depend on Q o V? Solution : No Q C. The constant C is called the capacitance of the V Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

18 PEPC 8 Q. Daw the gaph fo the vaiation of capacitance with chage on the capacito o potential diffeence acoss the capacito. Solution : Q. On what factos does the capacitance of capacito depend? Solution : The capacitance C depends only on the geometical configuation (shape, size, sepaation) of the system of two conductos, it also depends on the natue of the insulato (dielectic) sepaating the two conductos. Q. What is the SI unit of capacitance? Solution : The SI unit of capacitance is faad (= coulomb volt ) o F = C V. Q. Define faad? Solution : faad (F) capacitance of a capacito is defined, when a chage of coulomb aises the potential diffeence by V. Q. Give the symbol of capacitance? Solution : Q. Give the symbol of vaiable capacitance? Solution : Q. What is the advantage of lage value of capacitance of a capacito? Solution : A capacito with lage capacitance can hold lage amount of chage at a elatively small potential diffeence. Q. How the chage of the capacito leakes away due to the eduction in insulating powe of the intevening medium? Solution : High potential diffeence between the conductos of a capacito implies stong electic field aound the conductos. A stong electic field can ionised the suounding ai and acceleate the chages so poduced to the oppositely chaged plates, theeby neutalizing the chages on the capacito plates, at least patly. This is the way the chage of the capacito leakes away due to the eduction in insulating powe of the intevening medium. Q. Define dielectic stength of a medium o mateial? Solution : The maximum electic field that a dielectic medium can withstand without beakdown (of its insulating popety) is called its dielectic stength. Q. Is thee a limit to the amount of chage that can be stoed on a given capacito? Explain. Solution : Yes, thee is a limit to the amount of chage that can be stoed on a given capacito. Fo a capacito to stoe a lage amount of chage without leaking, its capacitance should be high enough so that the potential diffeence and hence the electic field do not exceed the beak-down limits. Q. What is the use of capacito? Solution : (i) Capacito stoes chages and hence electic enegy within it. (ii) It is a key element of most A.C. cicuit. (iii) Capacito ae used in many electical appliances fo e.g., oscillato cicuit, fan, motos etc.. The Paallel Plate Capacito : Q. What is paallel plate capacito? Deive the expession fo the capacitance of paallel plate capacito? Solution : A paallel plate capacito consists of two lage paallel conducting plates sepaated by a small distance. Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

19 PEPC 9 We fist take the intevening medium between the plates to be vacuum. Let A be the aea of each plate and d the sepaation between them. The two plates have chages Q and Q. Since d is much smalle than the linea dimension of the field by an infinite plane sheet of unifom suface chage density. Plate has a suface chage density = Q/A and plate has a suface chage density. In the inne egion between the plates and, the electic fields due to the two chaged plates add up, Q giving E. A Now fo unifom electic field, potential diffeence is simply the electic field times the distance between Qd the plates, that is, V Ed. A Q A The capacitance C of the paallel plate capacito is then C. V d Q. What is the natue of electic field inside the paallel plate capacito? Solution : Unifom. Q. Daw the gaph fo vaiation of electic field with distance fom positive plate to negative plate in paallel plate capacito. Solution : Q. Daw the gaph fo vaiation of electic potential with distance fom positive plate to negative plate in paallel plate capacito. Solution : Q. At one point in paallel plate capacito the electic field is E and at anothe point the field is E. What is the atio E : E? Solution : : Q. What is finging of the electic field in paallel plate capacito? Solution : Fo plates with finite aea the field lines bend outwad at the edges an effect called finging of the field. Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

20 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857 PEPC Q. Comment on the following statement : The capacitance F of a capacito is too big a unit in pactice. Solution : The bigness of F is to calculate the aea of the plates needed to have C = F fo a sepaation Cd F m 9 of, say cm : A m, which is a plate about 3 km in length and 8.85 C N m beadth!.3 Effect Of Dielectic On Capacitance : Q. How the capacitance of a paallel plate capacito is modified when a dielectic is intoduced between the plates? Explain. Solution : If a dielectic is inseted between the plates fully occupying the intevening egion then the dielectic is polaised by the field, the effect is equivalent to two chaged sheets (at the suface of the dielectic nomal to the field) with suface chage densities p and p. The electic field in the dielectic then coesponds to the case when the net suface chage density on the plates is ±( p ). That is P P E, so that the potential diffeence acoss the plates is V Ed d. Fo linea dielectics, we expect p to be popotional to E, i.e., to. Thus, ( p ) is popotional to and we can wite d have V K P Qd A, whee K is a constant chaacteistic of the dielectic. Clealy, K >. We then K Q KA. The capacitance C, with dielectic between the plates, is then C. K V d Q. Define dielectic constant in tems of capacitance of capacitos? Solution : C K. C The dielectic constant of a substance is the facto (>) by which the capacitance inceases fom its vacuum value, when the dielectic is inseted fully between the plates of a capacito. Q. A slab of mateial of dielectic constant K has the same aea as the plates of a paallel-plate capacito but has a thickness (3/4)d, whee d is the sepaation of the plates. How is the capacitance changed when the slab is inseted between the plates? [NCERT Solved Example.8] 4K Solution : C K 3.4 Combination Of Capacitance : Q. Conside two capacitos of capacitance C and C ae combined in seies combination. Deive the expession fo the effective capacitance? Solution : Figues shows capacitos C and C combined in seies. The left plate of C and ight plate of C ae connected to two teminals of a battey and have chages Q and Q, espectively. It then follows that the ight plate of C has chage Q and the left plate of C has chage Q. If this was not so, the net chage on each capacito would not be zeo. This would esult in an electic field in the conducto connecting C and C. Chage would flow until the net chage on both C and C is zeo and thee is no electic field in the conducto connecting C and C. Thus, in the seies combination,

21 PEPC chages on the two plates (+Q) ae the same on each capacito. The total potential dop V acoss the combination is the sum of the potential dops V and V acoss C and C, espectively. V V V Q C Q C i.e., V Q C C Now we can agad the combination as an effective capacito with chage Q and potential diffeence V. The effective capacitance of the combination is Q C. Hence we obtain V C Q. Conside two capacitos of capacitance C and C ae combined in paallel combination. Deive the expession fo the effective capacitance? Solution : Figue shows two capacitos aanged in paallel. C C In this case, the same potential diffeence is applied acoss both the capacitos. But the plate chages (+Q ) on capacito and the plate chages (+Q ) on the capacito ae not necessaily the same : Q = C V, Q = C V The equivalent capacito is one with chage Q = Q + Q and potential diffeence V. Hence Q = CV = C V + C V. The effective capacitance C is C = C + C Q. A netwok of fou µf capacitos is connected to a 5 V supply, as shown. Detemine (a) the equivalent capacitance of the netwok and (b) the chage on each capacito. (Note, the chage on a capacito is the chage on the plate with highe potential, equal and opposite to the chage on the plate with lowe potential). [NCERT Solved Example.9] Solution : (a) 3.3 µf (b) 5. 3 C Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

22 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , Enegy Stoed In A Capacito : Q. Deive the expession fo enegy stoed in a capacito? PEPC Solution : To detemine the enegy stoed in this configuation, conside initially two unchaged conductos and. Imagine next a pocess of tansfeing chage fom conducto to conducto bit by bit, so that at the end, conducto gets chage Q. By chage consevation, conducto has chage Q at the end. In tansfeing positive chage fom conducto to conducto, wok will be done extenally, since at any stage conducto is at a highe potential than conducto. To calculate the total wok done, we fist calculate the wok done in a small step involving tansfe of an infinitesimal (i.e., vanishingly small) amount of chage. Conside the intemediate situation when the conductos and have chages Q and Q espectively. At this stage, the potential diffeence V between conductos to is Q / C, whee C is the capacitance of the system. Next imagine that a small chage Q is tansfeed fom conducto to. Wok done in this step ( dw ), esulting in chage Q on conducto inceasing to Q dq Q dw VdQ dq C, is given by Total wok done equals to the enegy stoed in capacito, which will be obtain by integation as Q Q W dq C C Q Q Q C We can wite the final esult, in diffeent ways Q W CV QV (Q = CV) C 3 Q. Show that electic field E itself is a souce of enegy with enegy density E Jm. Solution : Since electostatic foce is consevative, this wok is stoed in the fom of potential enegy of the system. To see this, conside fo simplicity, a paallel plate capacito of aea A (of each plate) and sepaation d between the plates. Enegy stoed in the capacito Q C (A) d A The suface chage density is elated to the electic field E between the plates, E Hence, we get enegy stoed in the capacito U = (/) E Ad Note that Ad is the volume of the egion between the plates (whee electic field alone exists). If we define enegy density as enegy stoed pe unit volume of space, then enegy density of electic field u = (/) E Though we deived above equation fo the case of a paallel plate capacito, the esult on enegy density of an electic field is, in fact, vey geneal and holds tue fo electic field due to any configuation of chages.

23 PEPC 3 Q. (a) A 9 pf capacito is chaged by V battey. How much electostatic enegy is stoed by the capacito? (b) The capacito is disconnected fom the battey and connected to anothe 9 pf capacito. What is the electostatic enegy stoed by the system? [NCERT Solved Example.] Solution : (a) 4.5 µj (b).5 µj.6 Van De Gaaff Geneato : Q. What is Van De Gaaff Geneato? Solution : This is a machine that can build up the voltage in the ode of million ( 6 ) volts. Q. What is the use of high voltages developed by Van De Gaaff Geneato? Solution : The high voltages and hence lage electic fields, poduced by this geneato, ae used to acceleate chaged paticles (electons, potons, ions) to high enegies needed fo expeiments to pobe the small scale stuctue of matte. Q. Explain the basic pinciple, constuction and woking of a Van de Gaaff geneato with the help of a diagam. Solution : Pinciple, suppose we have a lage spheical conducting shell of adius R, on which we place a chage Q. This chage speads itself unifomly all ove the sphee. As we know, the field outside the sphee is just that of a point chage Q at the cente; while the field inside the sphee vanishes. So the potential outside is that of a point chage; and inside it is constant. We thus have : Potential inside conducting spheical shell of adius R caying chage Q = constant = 4 Now, as shown, let us suppose that in some way we intoduce a small sphee of adius, caying some chage q, into the lage one, and place it at the cente. The potential due to this new chage clealy has the following values at the adii indicated : Q R. Taking both chages q and Q into account, the potential of oute sphee is V(R) and potential of inne sphee V() is given by V(R) 4 Q R q R Q q, V() 4 R Hence potential diffeence between them equals to, q V() V(R) 4 Assume now that q is positive. We see that, independent of the amount of chage Q that may have accumulated on the lage sphee and even if it is positive, the inne sphee is always at a highe potential: the diffeence V() V(R) is positive. This means that if we now connect the smalle and lage sphee by a wie, the chage q will immediately flow to lage sphee. The natual tendency is fo positive chage to move fom highe to lowe potential. Thus, povided we ae somehow able to intoduced the small chaged sphee into the lage one, we can in this way keep piling up lage and lage amount of chage on the latte. The potential at the oute sphee would also keep ising, at least until we each the beakdown field of ai. R Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857

24 Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857 Constuction PEPC 4 A schematic diagam of the Van de Gaaff geneato is given in figue. A lage spheical conducting shell (of few metes adius) is suppoted at a height seveal metes above the gound on an insulating column. A long naow endless belt of insulating mateial, like ubbe o silk, is wound aound two pulleys - one at gound level, one at the cente of the shell. This belt is kept continuously moving by a moto diving the lowe pulley. It continuously caies positive chage, spayed on ot it by a bush at gound level, to the top. Thee it tansfes its positive chage to anothe conducting bush connected to the lage shell. Thus positive chage is tansfeed to the shell, whee it speads out unifomly on the oute suface. In this way, voltage diffeences of as much as 6 to 8 million volts (with espect to gound) can be built up.

25 PEPC 5 NCERT EXERCISE. Two chages 5 8 C and 3 8 C ae located 6 cm apat. At what point(s) on the line joining the two chages is the electic potential zeo? Take the potential at infinity to be zeo.. A egula hexagon of side cm has a chage 5 µc at each of its vetices. Calculate the potential at the cente of the hexagon..3 Two chages µc and µc ae placed at points A and B 6 cm apat. (a) Identify an equipotential suface of the system. (b) What is the diection of the electic field at evey point on this suface?.4 A spheical conducto of adius cm has a chage of.6 7 C distibuted unifomly on its suface. What is the electic field. (a) (b) inside the sphee just outside the sphee (c) at a point 8 cm fom the cente of the sphee?.5 A paallel plate capacito with ai between the plates has a capacitance of 8 pf ( pf = F). What will be the capacitance if the distance between the plates is educed by half, and the space between them is filled with a substance of dielectic constant 6?.6 Thee capacitos each of capacitance 9 pf ae connected in seies. (a) What is the total capacitance of the combination? (b) What is the potential diffeence acoss each capacito if the combination is connected to a V supply?.7 Thee capacitos of capacitances pf, 3 pf and 4 pf ae connected in paallel. (a) What is the total capacitance of the combination? (b) Detemine the chage on each capacito if the combination is connected to a V supply..8 In a paallel plate capacito with ai between the plates, each plate has an aea of 6 3 m and the distance between the plates is 3 mm. Calculate the capacitance of the capacito. If this capacito is connected to a V supply, what is the chage on each plate of the capacito?.9 Explain what would happen if in the capacito in the above poblem, a 3 mm thick mica sheet (of dielectic constant = 6) wee inseted between the plates. (i) while the voltage supply emained connected.. A pf capacito is connected to a 5V battey. How much electostatic enegy is stoed in the capacito?. A 6pF capacito is chaged by a V supply. It is then disconnected fom the supply and is connected to anothe unchaged 6 pf capacito. How much electostatic enegy is lost in the pocess? (ii) afte the supply was disconnected. ADDITIONAL EXERCISES. A chage of 8 mc is located at the oigin. Calculate the wok done in taking a small chage of 9 C fom a point P(,, 3 cm) to a point Q(, 4 cm, ) via a point R(, 6 cm, 9 cm)..3 A cube of side b has a chage q at each of its vetices. Detemine the potential and electic field due to this chage aay at the cente of the cube..4 Two tiny sphees caying chages.5 µc and.5 µc ae located 3 cm apat. Find the potential and electic field : (a) (b) at the mid-point of the line joining the two chages, and at a point cm fom this mid-point in a plane nomal to the line and passing though the mid-point..5 A spheical conducting shell of inne adius and oute adius has a chage Q. (a) (b) A chage q is placed at the cente of the shell. What is the suface chage density on the inne and oute suface of the shell? Is the electic field inside a cavity (with no chage) zeo, even if the shell is not spheical, but has any iegula shape? Explain. Einstein Classes, Unit No., 3, Vadhman Ring Road Plaza, Vikas Pui Extn., Oute Ring Road New Delhi 8, Ph. : , 857