CHAPTER 0 ELECTRIC POTENTIAL AND CAPACITANCE
ELECTRIC POTENTIAL AND CAPACITANCE 7 0. ELECTRIC POTENTIAL ENERGY Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by FE. When the paticle moves fom initial point i to final point f this foce does wok on it accoding to W f E dl 0. i whee dl is an infinitesimal displacement along an abitay path fom point i to point f. In section 9.3 we mentioned that the electostatic foce is consevative and so we can associate a potential enegy with this foce. The wok done by a consevative foce is defined to be eual to the negative of the change in the potential enegy, that is o U W 0. U f U i f i E dl 0.3 Let us now define the electic potential as the electic potential enegy pe unit chage, i.e., U 0.4
ELECTRIC POTENTIAL AND CAPACITANCE 8 The electic potential diffeence between the points i and f is then follows fom Euation 0.3 as f i f E dl 0.5 i Euation 0.5 admits the following definition: The potential diffeence f - i is defined as the wok euied to move a positive unit chage fom point i to point f. Since it is only the change in potential between two points that has physical sense, it is often convenient to choose a efeence point fo zeo potential. This efeence point is usually chosen to be at infinity, i.e., 0. Letting the point i to be and designate point f as point p Euation 0.5 becomes p p E dl 0.6 This gives the electic potential at any point p which can be defined as the wok euied to bing a positive unit chage fom infinity to that point. In this sense the potential at a point is the potential diffeence between that point and a point at infinity. As it is clea fom the definition given by Euation 0.4, the electic potential is a scala uantity. Theefoe, dealing with electic potential is easie than dealing with electic field which is
ELECTRIC POTENTIAL AND CAPACITANCE 9 vecto. Also Euation 0.4 tells that the SI unit of is joules pe coulomb (J/C), usually epesented by a special unit called volt () afte Alessando olta(745-87), i.e., J/C It follows fom Euation 0.5 that the electic field has a unit of volt pe mete (/m) with N/C /m Example 0. A unifom electic field E is diected along the x- axis as shown in Figue 0.. Calculate the potential diffeence between two points sepaated by a distance d, whee d is measued a long the diection of E. Solution The two points ae labeled A and B in the figue. Applying Euation 0.5 we E A B get d B A B B E dl Edl A A Figue 0. Example 0.. This follows fom the fact that E and dl ae paallel. Since E is unifom it can be taken out fom the integal sign, giving B E dl Ed A Fom the minus sign we conclude that A > B. Example 0. A poton is eleased fom est in a unifom electic field of.0 0 5 N/c diected along the positive x-axis, as shown in Figue 0.. Find the speed of the poton afte it has been displaced by 5 cm. v i 0 5 cm v f E Figue 0. Example 0..
ELECTRIC POTENTIAL AND CAPACITANCE 0 Solution Since the only foce acting on the poton is the electostatic foce, we can apply the pinciple of consevation of mechanical enegy in the fom K U 0, with U. o v 0 e ( ) 0 m p Solving fo v we get v e m p To find we use B A f E dl Edl i Note that the displacement of the positive poton is in the diection of E. So we have f E dl i.0 0 And fo the speed we obtain 5 f i (0.5) 5.0 0 9 4 (.6 0 )(5.0 0 ) 3. 0 m/s 6 7.67 0 4
ELECTRIC POTENTIAL AND CAPACITANCE 0. ELECTRIC POTENTIAL DUE TO POINT CHARGES Conside an isolated point chage. To find the electic potential due to this chage at a point, we apply Euation 0.5 along an abitay path as shown in Figue 0.3. If dl is an element along the path and a distance fom, the potential diffeence between any two points A and B along the path is A B dl Figue 0.3 The electic potential at appoint a distance fom a point chage. A B B A B E dl 0.7 A The electic field due to the chage at a distance is E k ˆ, whee ˆ is a unit vecto diected adially outwad fom. Theefoe, Euation 0.7 becomes B A b ˆ dl k a 0.8
ELECTRIC POTENTIAL AND CAPACITANCE But ˆ dl gives the pojection of dl along the adial diection, i.e., ˆ d l d Substituting back into Euation 0.7 and integating we get B A k b a k b k a 0.9 The fist tem of the ight hand side of Euation 0.8 epesents the potential at point B ( B ) and the second tem epesents the potential at point A ( A ). The electic potential at a point a distance fom a point chage is then obtained if a in Euation 0.9, that is, k 0.0 As it is clea fom the above euation, is positive fo positive and negative fo negative. The electic potential due to a goup of point chages at a point is the algebaic sum of the electic potentials due to each chage individually. That is, fo a goup of N point chages we have k N i i i 0.
ELECTRIC POTENTIAL AND CAPACITANCE 3 whee i is the distance fom the ith chage i to the point in uestion. Do not foget that the sum in Euation 0. is algebaic sum and not vecto sum like that used to calculate the electic filed due to a goup of point chages. This fact gives an impotant advantage of potential ove electic field. To find the electic potential at a point outside a spheical shell we again teat such a shell as if all its chages ae concentates at its cente. This implies that such a potential will be given by Euation 0.0 but now epesents the distance between the point and the cente of the shell. Example 0.3 Thee point chages of -4.0 µc, µc, and 3 6 µc ae aanged as shown in Figue 0.4. Find the electic potential at the point P. Solution Fom Euation 0. we wite fo N 3 k 3 3 As it is clea fom the figue, the µ Figue 0.4 Example 0.3. distances fom each chage to the point p ae 4.0 m, 4.0 m, and 3 6.0 m. Substituting fo these values and the values of the chages in the above fomula we get 9 4.0 9 0 4.0 4 4. 0 4.0 6 6.0-4.0 µc 0 4.0 m 6 3 6 µc 6.0 m 4.0 m P µc
ELECTRIC POTENTIAL AND CAPACITANCE 4 Note that the negative sign of is included in ou calculation. 0.3 POTENTIAL ENERGY OF A SYSTEM OF POINT CHARGES We now want to calculate the electostatic enegy associated with a system of point chages. By this we mean the wok euied to 3 assemble this system fom a condition 3 whee all chages ae infinitely sepaated fom one anothe. To do so 3 we have to calculate the wok euied to bing, fom infinity, each chage one by one. Figue 0.5 As defined in section 0., the potential at a point is eual to the wok euied to bing a positive unit chage fom infinity to that point. Theefoe, the wok euied to bing a chage fom infinity to a point must eual to the potential at that point multiplied by, i.e., W The potential enegy of a system of thee-point chages. p p 0. The minus sign of Euation 0. is dopped hee because we ae speaking with the wok done by an extenal agent athe that the wok done by the field. Suppose that we want to assemble a system of thee point chages as shown in Figue 0.5. Accoding to Euation 0., no
ELECTRIC POTENTIAL AND CAPACITANCE 5 wok is euied to place the fist chage at a given position because thee is no electic potential at that position. Next we place a second chage at a position fom. This euies a wok whee is the potential at the location of due to. If we denote this wok by W with W 0 denotes the wok euied to place, then we have W k Next we place 3 at a position 3 fom and 3 fom. We now must do wok given as 3, whee is the potential at the location of 3 due both and, i.e., W 3 k 3 3 3 The total wok euied to assemble a system of thee chages is then W W W W3 This wok is stoed as an electostatic enegy in the system, so we wite 3 3 U W k 0.3 3 3
ELECTRIC POTENTIAL AND CAPACITANCE 6 The potential enegy of a system of N chages can be calculated in a simila fashion. The esult can be witten as U j j k N j k jk k 0.4 Example 0.4 Two point chages of P 6.0 µc, and 8.0 µc, ae located along the x-axis m apat, as 3.0 m shown in Figue 0.6..0 m.0 m a) What is the wok euied to bing 6.0 µc 8.0 µc a thid chage 3 -.0 µc fom Figue 0.6 Example 0.4. infinity to the point p. b) Calculate the electostatic potential enegy of the thee chages system. Solution a) Fom Euation 0., the wok euied to bing 3 fom infinity to point P is simple the potential at point P multiplied by the chage of 3. The potential at p is p k 9 0 6 6.0 0 8.0 0 3. 3. 3.9 6 9 0 4 So the wok euied is
ELECTRIC POTENTIAL AND CAPACITANCE 7 W 3 P 6 4 (.0 0 )( 3.9 0 ) 7.8 0 J b) Using Euation 0.3 we have U k 3 3 3 3 9 0 9 (6.0)(8.0) 0 (6.0)(.0) 0 (8.0)(.0).0 3. 3..4 0 J 0.4 ELECTRIC POTENTIAL AND CONDUCTORS Conductos in electostatic euilibium have vey impotant popeties mentioned in section 9.8. Anothe impotant popety concening the electic potential is The electic potential inside any conducto is constant and eual to the potential on its suface. To pove this popety we have f i f E dl i If the initial and the final points lie inside a conducto, and using the fact that E0 inside a conducto, we conclude that the electic
ELECTRIC POTENTIAL AND CAPACITANCE 8 potential is constant inside the conducto. Futhemoe, since E is always nomal to dl on the suface, the electic potential is also constant on the suface of the conducto. Conside again a conducting sphee of adius R and chage. The electic potential at a point inside the sphee is, fom Euation 0.6 E d R E out d E R in d But E in 0 and E out K ˆ, so we get K R which is the value of the potential at the suface of the conducto. If two o moe conducting objects ae connected by a conducting wie, the conductos ae no longe sepaate but can be consideed as a single conducto. This means that the electic chages will tansfe fom the conducto of highe potential to that of lowe potential until the euilibium condition is achieved. Theefoe, if two o moe conductos ae connected and euilibium is achieved, they must be at the same electic potential. In analogue with the electic filed lines, the electic potential can be epesented by euipotential sufaces. A suface with all its points ae at the same electic potential is called
ELECTRIC POTENTIAL AND CAPACITANCE 9 euipotential suface. Fom this definition, it follows that no wok is done by the electic field in moving a chaged paticle between tow points on the same euipotential suface. This means that the electic filed lines must be pependicula to the euipotential sufaces. Figue 0.- shows the euipotential sufaces fo some common chage distibutions. The suface of any conducto foms an euipotential suface. As a conseuence of this fact, chage tends to accumulate at shap points, as will be poved in example 0.8. Example 0.8 Two conducting chaged sphees with adii R and R ae sepaated by a distance much lage than the adius of eithe sphee. The two sphees ae connected by a conducting wie. a) Find the atio of the final chages on the sphees. b) Find the atio of the chage densities on the sufaces of the sphees. Solution Since the sphees ae connected by a conducting wie, the potential is the same fo both sphees, i.e., K R K R Fom which it follows that R R
ELECTRIC POTENTIAL AND CAPACITANCE 30 b) The chages ae distibuted ove the suface of the sphees, so we have Theefoe σ and 4πR σ 4πR σ σ Substituting fo the atio given above we get σ σ It is clea fom this esult that the chage density is geatest on the small sphee as expected. Futhemoe, since the electic field just outside a conducto is popotional to the chage density, the field is moe intense nea the smalle sphee. 0.5 CAPACITANCE Any two conductos with an insulato between them fom a capacito. The conductos usually have eual but opposite chages so that the net chage of any capacito is zeo. Howeve, when we said that a capacito has a chage, we mean that one of
ELECTRIC POTENTIAL AND CAPACITANCE 3 the conductos has a chage and the othe has a chage -. The capacitance C of a capacito is defined as the atio of the magnitude of the chage on eithe conductos to the magnitude of the potential diffeence, denoted heeafte by, i.e., C 0.5 The capacitance is always a positive uantity. Fom Euation 4. it is clea that the SI unit of the capacitance C is Coulomb pe volt (C/). This unit is efeed as Faad (F) in hono of Michael Faaday, that is F C/ Capacitos ae vey vital elements of almost all electonic devices used to stoe chages and conseuently electostatic enegy. They ae essential in cicuits we use to tune adio and television tansmittes and eceives. They ae used also to egulate the output of the electonic powe supplies. The micoscopic capacitos fom the memoy bank of computes. The Paallel Plate Capacito The capacito consists of two paallel metal plates of eual aea A sepaated by a distance d and immesed in vacuum, as shown in Figue Figue 0.8 A paallel plate capacito with plate's aea A and sepaation d.
ELECTRIC POTENTIAL AND CAPACITANCE 3 0.8. The two plates have eual but opposite chages, and -. If the sepaation d is small compaed to the size of the plates we can assume the electic field to be unifom in the egion between the plates. In this egion any point can be consideed to be just outside a conducto, and so the magnitude of the electic field of such a configuation, fom Euation 9.6, is E σ ε o with σ A is the chage density on eithe plate. Using Euation 0.5 with the integal is to be evaluated along a path that stats at one plate and ends on the othe, i.e., E dl whee and efe, espectively to the positive and negative plates of the capacito. Since E is constant and diected fom the positive to the negative plate we can wite E dl Ed σ d ε o Aε o d Substituting this value into Euation 4. we obtain
ELECTRIC POTENTIAL AND CAPACITANCE 33 o C d Aε o C A ε o 0.6 d This means that the capacitance depends only on the geomety of the capacito; diectly popotional to the aea of the plates and invesely popotional to the sepaation with the pemittivity ε o stands fo the popotionality constant. We will see late that if the medium between the plates is not vacuum this constant should be multiplied by a facto. In cicuits the capacitos and the batteies ae epesented as shown in Figue 0.9. We can chage a capacito by connecting it acoss the teminal of a battey as shown in Figue 0.0. In doing so electons tansfe fom the plate that is connected to the positive teminal of the battey ( the longe line of the battey symbol) to the plate that is connected to the negative teminal of the battey. This pocess continues fo a shot time until the potential diffeence acoss the capacito becomes eual to the potential diffeence of the battey.
ELECTRIC POTENTIAL AND CAPACITANCE 34 Example 0.9 A paallel plate capacito of sepaation.00 mm and plate aea of 0.500 m ae connected to a 800.- battey. Calculate (a) the capacitance of the capacito, (b) the chage on the capacito Solution (a) Using Euation 4. we have A 0.500 C ε 8.85 0 3 d.00 0 (b) Fom Euation 4. we get o.pf C 9 (. 0 )( 800).77µ C 0.6 COMBINATIONS OF CAPACITORS It occus sometimes that a goup of capacitos ae connected in the same cicuit. The combination is often eplaced by a single euivalent capacito with a capacitance eual to the capacitance of Capacito symbol - Battey symbol Figue 0.9 Cicuit symbols fo capacitos and batteies. Figue 0.0 Capacito chaged to a battey.
ELECTRIC POTENTIAL AND CAPACITANCE 35 the whole combination. Two basic combinations of capacitos ae discussed hee: The paallel, and the seies combinations. I-Paallel Combination If two capacitos ae connected as shown in Figue 0. we say that the two capacitos ae connected in paallel. In such a combination, the ight plates ae connected, by a conducting wie, togethe to fom an euipotential suface (They have the same potential). The othe two plates, the left plates, fom a anothe euipotential suface. Theefoe, the potential diffeence acoss the two capacitos ae the same and eual to the potential diffeence acoss an euivalent capacito eplacing the two capacitos, then we have 0.7 e C C Figue 0. Two capacitos ae connected in paallel. whee e stands fo the potential diffeence acoss the euivalent capacito. The chage stoed by this euivalent capacito e is eual to the sum of the chages stoed by each capacito, i.e., 0.8 e Using Euation 0.5, the last euation can be witten as C C C e e
ELECTRIC POTENTIAL AND CAPACITANCE 36 Using Euation 0.7 we finally obtain C C C 0.9 e This can be extend to any numbe of capacitos, that is Capacitos ae said to be connected in paallel if the potential acoss each one is the same and eual to the potential acoss an euivalent capacito. The euivalent capacitance of the combination is eual to the sum of the capacitance of each capacito. II Seies Combination Figue 0. illustates two capacitos connected in seies. When the battey is connected, electons tansfe fom the left plate of C to the ight plate of C though the battey. Thus, the left plate of C acuies a positive chage while the ight plate of C acuies an eual negative chage. As the othe two plates, enclosed by the dashed line in Figue 0., fom an isolated conducto, electons ae attacted to the left end leaving the ight end with an excess C C Figue 0. Two capacitos ae connected in seies. positive chages. This means that the battey induces a chage on the isolated conducto. It is clea hee that the chages on each
ELECTRIC POTENTIAL AND CAPACITANCE 37 capacito must be the same and eual to the chage on an euivalent capacito eplacing the two capacitos, that is, 0.0 e and 0. ab e Again using Euation 0.5 we wite C e e C C But, fom Euation 0.0, the nominatos cancel yielding C e 0. C C This also can be genealized to any numbe of capacitos connected in seies. Thus Capacitos ae said to be connected in seies if the chage on each one is the same and eual to the chage on an euivalent capacito. The ecipocal of the euivalent capacitance euals the sum of the ecipocals of the capacitance of each capacito.
ELECTRIC POTENTIAL AND CAPACITANCE 38 C C C C e C 3 C (a) (b) (c) Figue 0.3 Example 0.0. Example 0.0 Thee capacitos, C 6.00µ F, C 6.00µ F, and C3 4.00µ Fae connected as shown in Figue 0.3. (a) What is the euivalent capacitance of the combination? (b) If the combination is connected to a battey of.0, calculate the potential diffeence acoss, and the chage on each capacito. Solution (a) Using the seies and paallel ules, we educe the combination step by step until we each a single capacito, consideed to be the euivalent capacito, as indicated in the figue. C and C ae in seies. Fom Euation 0., thei euivalent capacitance is CC C 3.00µ F C C As Figue 0.3b shows, C and C 3 ae connected in paallel. Thei euivalent capacitance, fom Euation 0.9 is
ELECTRIC POTENTIAL AND CAPACITANCE 39 C C C C 3.00 4.00 7.00µ F e 3 3 (b) Since C and C 3 ae connected in paallel then 3 e.0 Now fom Euation 0.5, we have and (.0)( 4.00) ) 48.0 C 3 C33 µ (.0)( 3.00) ) 36.0 C C µ This same chage exists on C and on C due to the seies combination between them, i.e., 36.0µ C To find the potential diffeence acoss C and C we have 36 6.0 36 6.0 C C 6.00, and 6.00 0.7 ENERGY STORED IN A CHARGED CAPACITOR
ELECTRIC POTENTIAL AND CAPACITANCE 40 As mentioned in the pevious section, in chaging a capacito electons ae tansfeed fom one plate to the othe building up a potential diffeence acoss the capacito. This means that a wok is euied to chage a capacito and this wok is stoed as a potential enegy in the capacito. To calculate the potential enegy U stoed in a chaged capacito, we conside a paallel plates capacito that is initially unchaged. Suppose that is the chage built up on the capacito at some instant duing the chaging pocess. The potential diffeence acoss the capacito at that instant is, fom Euation 0.5, v C, with C is the capacitance of the capacito. The wok euied to tansfe a small chage d is theefoe dw vd C d The total wok euied to chage a capacito fom unchaged situation (0) to a final chage is, thus W d 0.3 C C 0 This total wok will be stoed in the capacito as potential enegy. Using Euation 0.5 we can wite U C 0.4 C
ELECTRIC POTENTIAL AND CAPACITANCE 4 Example 0. Refeing to the pevious example Find (a) The enegy stoed by each capacito. (b) The enegy stoed by the goup. Solution (a) The chage on each capacito if found in the pevious example. Using Euation 0.4 we have U U ( 36.0) C.0 ( 36.0) C.0 08 J 08J U ( 48.0) 3 3 C 3 8.00 88J (c) Using again Euation 0.4 we have U C ( 7.00.0) e e e 7.00 Note that U e U U U 3 U 4 504 J 0.8 CAPACITORS WITH DIELECTRIC
ELECTRIC POTENTIAL AND CAPACITANCE 4 A dielectic is an insulating mateial such as ubbe, glass, o plastic. It is found that when a dielectic mateial is inseted between the plates of a capacito, its capacitance inceases by a numeical facto κ called the dielectic constant of the mateial, that is C κc o 0.5 whee C and C o ae the capacitance with and without the dielectic espectively. Since C is always geate than C o, the dielectic constant κ must be geate than unity. Table 0. gives the dielectic constants fo some mateials. Let us now explain what happen when a dielectic mateial is inseted between the plates of an isolated capacito. Suppose that the dielectic is a pola mateial (has a pemanent electic dipole). The electic field of the capacito E o exets a toue on the dipoles of the mateial so that it tends to otates Table 0. Dielectic constants and dielectic stengths fo some mateials at oom tempeatue. The dielectic stength is the maximum electic field befoe beakdown (chage E o flow) occus. E o - - Dielectic Dielectic - - - Mateial constant stength - - (/m) - E' - acuum -.0000 3 0 6 - Ai (dy).0006 4 0 6 - - - Paaffin -. 0 0 6 Polystyene -.6 4 0 6 (a) (b) Pape 3.7 (c) 5 0 6 uatz 4.3 8 0 6 Figue 0.4 (a) A dielectic slab with its molecules ae andomly oiented. (b) A chaged capacito without a dielectic. E Oil 4 0 6 o is the electic field inside the capacito without the dielectic. (c) The dielectic being Glass inseted between 5 the chaged 4 0 6 capacito. The net electic field is now E o -E.' Rubbe 6.7 0 6 Pocelain 6-8 5 0 6 Nylon 3.4 4 0 6 Wate 80
ELECTRIC POTENTIAL AND CAPACITANCE 43 these dipoles into the diection of E o. In this case the mateial is said to be polaized. As a esult of this polaization, polaization chages ae poduced at the faces of the dielectic as shown in Figue 0.4.(the chages inside the dielectic cancel each othe). Note that the positive chages ae nea the negative plate and the negative chages ae nea the positive plate. These chages ceate an electic field E' opposite to E o. The net electic filed inside the conducto is theefoe E E o E 0.6 That is, a eduction is occued in the electic field. If the dielectic is not pola, it will acuies a nonpemanent, induced dipole moment when placed in an extenal electic field. The effect of the extenal electic field is to stetch the molecule and theefoe, the centes of the positive and the negative chages ae slightly sepaated. These induced dipoles tend to align with the extenal electic field so that polaization chages fomed at the faces of the dielectic and so a eduction of the electic field is also achieved. Since Ed, the potential diffeence acoss the capacito deceases, conseuently. Now fom Euation 0.5 and since the chage is constant, we conclude that the capacitance is inceased by insetion of a dielectic mateial between the conductos of a capacito Example 0. A paallel plate capacito, of capacitance C 80 o. µ F is chaged by a 36- battey. The battey is then emoved and a dielectic slab of dielectic constant κ 3. 7 is inseted between the plates of the capacito.
ELECTRIC POTENTIAL AND CAPACITANCE 44 a) What is the capacitance afte inseting the dielectic? b) Find the enegy stoed in the capacito befoe and afte the slab is inseted. Solution (a) Fom Euation 0.5 we have C κ Co (3.6)(8.0µ F) 9µ F (b) The enegy stoed in the capacitance befoe inseting the dielectic is U o o o C µ 3 ( 8.0 F)( 36) 5. 0 J Afte emoving the battey the chage on the capacito will not be changed even if the capacitance is inceased by inseting the dielectic. This unchanged chage is ( 36)( 8.0µ F) 90 C o ε Co µ Now, the enegy stoed afte intoducing the dielectic is o U C 6 ( 88 0 ) 6 ( 8.8 0 ) The diffeence in the enegy done by the capacito. U o.4 0 3 J U can be explained as the wok