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www.GOALias.blogspot.comPhysicsFIGURE 2.13 Potential energy of asystem of charges q 1and q 2isdirectly proportional to the productof charges and inversely to thedistance between them.where r 12is the distance between points 1 and 2.Since electrostatic force is conservative, this work getsstored in the form of potential energy of the system. Thus,the potential energy of a system of two charges q 1and q 2isU1qq= 1 24 π ε0 r(2.22)12Obviously, if q 2was brought first to its present location andq 1brought later, the potential energy U would be the same.More generally, the potential energy expression,Eq. (2.22), is unaltered whatever way the charges are brought to the specifiedlocations, because of path-independence of work for electrostatic force.Equation (2.22) is true for any sign of q 1and q 2. If q 1q 2> 0, potentialenergy is positive. This is as expected, since for like charges (q 1q 2> 0),electrostatic force is repulsive and a positive amount of work is needed tobe done against this force to bring the charges from infinity to a finitedistance apart. For unlike charges (q 1q 2< 0), the electrostatic force isattractive. In that case, a positive amount of work is needed against thisforce to take the charges from the given location to infinity. In other words,a negative amount of work is needed for the reverse path (from infinity tothe present locations), so the potential energy is negative.Equation (2.22) is easily generalised for a system of any number ofpoint charges. Let us calculate the potential energy of a system of threecharges q 1, q 2 and q 3 located at r 1 , r 2 , r 3 , respectively. To bring q 1 firstfrom infinity to r 1, no work is required. Next we bring q 2from infinity tor 2. As before, work done in this step isqV1 qq= πε1 22 1( r 2)4 0r(2.23)12The charges q 1and q 2produce a potential, which at any point P isgiven by1 ⎛ q1 q2⎞V1, 2=4 ε ⎜ +0r1P r ⎟(2.24)π ⎝ 2P⎠Work done next in bringing q 3from infinity to the point r 3is q 3timesV 1, 2at r 31 qq1 3qq2 3qV3 1,2( r ⎛⎞3)= +4πε⎜0 ⎝ r13 r ⎟23 ⎠(2.25)The total work done in assembling the chargesat the given locations is obtained by adding the workdone in different steps [Eq. (2.23) and Eq. (2.25)],FIGURE 2.14 Potential energy of asystem of three charges is given byEq. (2.26), with the notation givenin the figure.621 ⎛qq1 2qq1 3qq2 3⎞U =4 ε ⎜ + +0r12 r13 r ⎟(2.26)π ⎝ 23 ⎠Again, because of the conservative nature of theelectrostatic force (or equivalently, the pathindependence of work done), the final expression forU, Eq. (2.26), is independent of the manner in whichthe configuration is assembled. The potential energy