Chapter Two ELECTROSTATIC POTENTIAL AND …

1 INTRODUCTIONIn Chapters 6 and 8 (Class XI), the notion of POTENTIAL energy wasintroduced. When an external force does work in taking a body from apoint to another against a force like spring force or gravitational force,that work gets stored as POTENTIAL energy of the body. When the externalforce is removed, the body moves, gaining kinetic energy and losingan equal amount of POTENTIAL energy. The sum of kinetic andpotential energies is thus conserved. Forces of this kind are calledconservative forces. Spring force and gravitational force are examples ofconservative force between two (stationary) charges is also a conservativeforce. This is not surprising, since both have inverse-square dependenceon distance and differ mainly in the proportionality constants themasses in the gravitational law are replaced by charges in Coulomb slaw. Thus, like the POTENTIAL energy of a mass in a gravitationalfield, we can define ELECTROSTATIC POTENTIAL energy of a charge in anelectrostatic an ELECTROSTATIC field E due to some charge , for simplicity, consider the field E due to a charge Q placed at theorigin.

2 Now, imagine that we bring a test charge q from a point R to apoint P against the repulsive force on it due to the charge Q. With referenceChapter TwoELECTROSTATICPOTENTIAL ANDCAPACITANCEP hysics52to Fig. , this will happen if Q and q are both positiveor both negative. For definiteness, let us take Q, q > remarks may be made here. First, we assumethat the test charge q is so small that it does not disturbthe original configuration, namely the charge Q at theorigin (or else, we keep Q fixed at the origin by someunspecified force). Second, in bringing the charge q fromR to P, we apply an external force Fext just enough tocounter the repulsive electric force FE ( , Fext= FE).This means there is no net force on or acceleration ofthe charge q when it is brought from R to P, , it isbrought with infinitesimally slow constant speed. Inthis situation, work done by the external force is the negative of the workdone by the electric force, and gets fully stored in the form of potentialenergy of the charge q.

3 If the external force is removed on reaching P, theelectric force will take the charge away from Q the stored energy (potentialenergy) at P is used to provide kinetic energy to the charge q in such away that the sum of the kinetic and POTENTIAL energies is , work done by external forces in moving a charge q from R to P isWRP = PRde xt Fri = PRdE Fri ( )This work done is against ELECTROSTATIC repulsive force and gets storedas POTENTIAL every point in electric field, a particle with charge q possesses acertain ELECTROSTATIC POTENTIAL energy, this work done increases its potentialenergy by an amount equal to POTENTIAL energy difference between pointsR and , POTENTIAL energy differencePRRPUUUW = =( )(Note here that this displacement is in an opposite sense to the electricforce and hence work done by electric field is negative, , WRP.)Therefore, we can define electric POTENTIAL energy difference betweentwo points as the work required to be done by an external force in moving(without accelerating) charge q from one point to another for electric fieldof any arbitrary charge important comments may be made at this stage:(i)The right side of Eq.

4 ( ) depends only on the initial and final positionsof the charge. It means that the work done by an ELECTROSTATIC field inmoving a charge from one point to another depends only on the initialand the final points and is independent of the path taken to go fromone point to the other. This is the fundamental characteristic of aconservative force. The concept of the POTENTIAL energy would not bemeaningful if the work depended on the path. The path-independenceof work done by an ELECTROSTATIC field can be proved using theCoulomb s law. We omit this proof A test charge q (> 0) ismoved from the point R to thepoint P against the repulsiveforce on it by the charge Q (> 0)placed at the Potentialand Capacitance53(ii)Equation ( ) defines POTENTIAL energy difference in termsof the physically meaningful quantity work. Clearly, POTENTIAL energy so defined is undetermined to within anadditive this means is that the actual valueof POTENTIAL energy is not physically significant; it is onlythe difference of POTENTIAL energy that is significant.

5 We canalways add an arbitrary constant to POTENTIAL energy atevery point, since this will not change the POTENTIAL energydifference:() ()PRPRUUUU + += Put it differently, there is a freedom in choosing the pointwhere POTENTIAL energy is zero. A convenient choice is to haveelectrostatic POTENTIAL energy zero at infinity. With this choice,if we take the point R at infinity, we get from Eq. ( )PPPWUUU = =( )Since the point P is arbitrary, Eq. ( ) provides us with adefinition of POTENTIAL energy of a charge q at any energy of charge q at a point (in the presence of fielddue to any charge configuration) is the work done by theexternal force (equal and opposite to the electric force) inbringing the charge q from infinity to that ELECTROSTATIC POTENTIALC onsider any general static charge configuration. We definepotential energy of a test charge q in terms of the work doneon the charge q. This work is obviously proportional to q, sincethe force at any point is qE, where E is the electric field at thatpoint due to the given charge configuration.

6 It is, therefore,convenient to divide the work by the amount of charge q, sothat the resulting quantity is independent of q. In other words,work done per unit test charge is characteristic of the electricfield associated with the charge configuration. This leads tothe idea of ELECTROSTATIC POTENTIAL V due to a given chargeconfiguration. From Eq. ( ), we get:Work done by external force in bringing a unit positivecharge from point R to P= VP VR PRUUq = ( )where VP and VR are the ELECTROSTATIC potentials at P and R, , as before, that it is not the actual value of POTENTIAL but the potentialdifference that is physically significant. If, as before, we choose thepotential to be zero at infinity, Eq. ( ) implies:Work done by an external force in bringing a unit positive chargefrom infinity to a point = ELECTROSTATIC POTENTIAL (V) at that ALESSANDRO VOLTA (1745 1827)Count Alessandro Volta(1745 1827) Italianphysicist, professor atPavia. Volta establishedthat the animal electri-city observed by LuigiGalvani, 1737 1798, inexperiments with frogmuscle tissue placed incontact with dissimilarmetals, was not due toany exceptional propertyof animal tissues butwas also generatedwhenever any wet bodywas sandwiched betweendissimilar metals.

7 Thisled him to develop thefirst voltaic pile, orbattery, consisting of alarge stack of moist disksof cardboard (electro-lyte) sandwichedbetween disks of metal(electrodes).Physics54In other words, the ELECTROSTATIC POTENTIAL (V)at any point in a region with ELECTROSTATIC field isthe work done in bringing a unit positivecharge (without acceleration) from infinity tothat qualifying remarks made earlier regardingpotential energy also apply to the definition ofpotential. To obtain the work done per unit testcharge, we should take an infinitesimal test charge q, obtain the work done W in bringing it frominfinity to the point and determine the ratio W/ q. Also, the external force at every point ofthe path is to be equal and opposite to theelectrostatic force on the test charge at that POTENTIAL DUE TO A POINT CHARGEC onsider a point charge Q at the origin (Fig. ). For definiteness, take Qto be positive. We wish to determine the POTENTIAL at any point P withposition vector r from the origin.

8 For that we mustcalculate the work done in bringing a unit positivetest charge from infinity to the point P. For Q > 0,the work done against the repulsive force on thetest charge is positive. Since work done isindependent of the path, we choose a convenientpath along the radial direction from infinity tothe point some intermediate point P on the path, theelectrostatic force on a unit positive charge is201 4'Qr r( )where ris the unit vector along OP . Work doneagainst this force from r to r + Dr is204'QWrr = ( )The negative sign appears because for Dr < 0, DW is positive . Totalwork done (W) by the external force is obtained by integrating Eq. ( )from r = to r = r,2000444'rrQQQW drrrr = == ( )This, by definition is the POTENTIAL at P due to the charge Q0( )4QV rr = ( )FIGURE Work done on a test charge qby the ELECTROSTATIC field due to any givencharge configuration is independentof the path, and depends only onits initial and final Work done in bringing a unitpositive test charge from infinity to thepoint P, against the repulsive force ofcharge Q (Q > 0), is the POTENTIAL at P due tothe charge Potentialand Capacitance55 EXAMPLE ( ) is true for anysign of the charge Q, though weconsidered Q > 0 in its Q < 0, V < 0, , work done (bythe external force) per unit positivetest charge in bringing it frominfinity to the point is negative.

9 Thisis equivalent to saying that workdone by the ELECTROSTATIC force inbringing the unit positive chargeform infinity to the point P ispositive. [This is as it should be,since for Q < 0, the force on a unitpositive test charge is attractive, sothat the ELECTROSTATIC force and thedisplacement (from infinity to P) arein the same direction.] Finally, wenote that Eq. ( ) is consistent withthe choice that POTENTIAL at infinitybe ( ) shows how the ELECTROSTATIC POTENTIAL ( 1/r) and theelectrostatic field ( 1/r2 ) varies with (a)Calculate the POTENTIAL at a point P due to a charge of 4 10 7 Clocated 9 cm away.(b)Hence obtain the work done in bringing a charge of 2 10 9 Cfrom infinity to the point P. Does the answer depend on the pathalong which the charge is brought?Solution(a) = 4 104 V(b)W = qV = 2 10 9C 4 104V = 8 10 5 JNo, work done will be path independent. Any arbitrary infinitesimalpath can be resolved into two perpendicular displacements: One alongr and another perpendicular to r.

10 The work done corresponding tothe later will be POTENTIAL DUE TO AN ELECTRIC DIPOLEAs we learnt in the last Chapter , an electric dipole consists of two chargesq and q separated by a (small) distance 2a. Its total charge is zero. It ischaracterised by a dipole moment vector p whose magnitude is q 2aand which points in the direction from q to q (Fig. ). We also saw thatthe electric field of a dipole at a point with position vector r depends notjust on the magnitude r, but also on the angle between r and p. Further,FIGURE Variation of POTENTIAL V with r [in units of(Q/4 0) m-1] (blue curve) and field with r [in unitsof (Q/4 0) m-2] (black curve) for a point charge field falls off, at large distance, not as1/r2 (typical of field due to a single charge)but as 1/r3. We, now, determine the electricpotential due to a dipole and contrast itwith the POTENTIAL due to a single before, we take the origin at thecentre of the dipole. Now we know that theelectric field obeys the superpositionprinciple.