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www.GOALias.blogspot.comPhysicsFIGURE 1.6 (a) Geometry and(b) Forces between charges.Since force is a vector, it is better to writeCoulomb’s law in the vector notation. Let theposition vectors of charges q 1and q 2be r 1and r 2respectively [see Fig.1.6(a)]. We denote force onq 1due to q 2by F 12and force on q 2due to q 1byF 21. The two point charges q 1and q 2have beennumbered 1 and 2 for convenience and the vectorleading from 1 to 2 is denoted by r 21:r 21= r 2– r 1In the same way, the vector leading from 2 to1 is denoted by r 12:r 12= r 1– r 2= – r 21The magnitude of the vectors r 21and r 12isdenoted by r 21and r 12, respectively (r = r ). The12 21direction of a vector is specified by a unit vectoralong the vector. To denote the direction from 1to 2 (or from 2 to 1), we define the unit vectors:ˆ21r21r,21= rrrˆ = , rˆ = rˆ1212 21 12r12Coulomb’s force law between two point charges q 1and q 2located atr 1and r 2is then expressed as1 q qF =r ˆ(1.3)1 221 2 214 π εor21Some remarks on Eq. (1.3) are relevant:• Equation (1.3) is valid for any sign of q 1and q 2whether positive ornegative. If q 1and q 2are of the same sign (either both positive or bothnegative), F 21is along ˆr 21, which denotes repulsion, as it should be forlike charges. If q 1and q 2are of opposite signs, F 21is along – ˆr 21(= ˆr 12),which denotes attraction, as expected for unlike charges. Thus, we donot have to write separate equations for the cases of like and unlikecharges. Equation (1.3) takes care of both cases correctly [Fig. 1.6(b)].• The force F 12on charge q 1due to charge q 2, is obtained from Eq. (1.3),by simply interchanging 1 and 2, i.e.,1 q qF = rˆ= −F1 212 2 12 214 π ε0 r1212Thus, Coulomb’s law agrees with the Newton’s third law.• Coulomb’s law [Eq. (1.3)] gives the force between two charges q 1andq 2in vacuum. If the charges are placed in matter or the interveningspace has matter, the situation gets complicated due to the presenceof charged constituents of matter. We shall consider electrostatics inmatter in the next chapter.