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Two charges -q and +q are locatedat points (0,0,-a) and (0,0,a) respectively. (i) What is the electrostaticpotentialat the pointsat the points (0,0,z) and (x,y,0)? (ii) Obtain the depentenceof potentialon the distancer of the point form theorigin, when (r)/(a) gt gt 1. (iii) How much work is done in movinga small test chargefrom the point (5,0,0) to (-7,0,0) alongthe x-axis. Does the answercharge if path of test chargebetweenthe same points is not along thex-axis ? |
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Answer» <P> Solution :Here, -q is at (0,0, -a) and +q is at (0,0,a)(i) Potential at `(0,0,z)` would be `V = (1)/(4pi in_(0)) ((-q)/(z+a)) + (1)/(4pi in_(0)) (q)/((z-a)) = (q(z+a-z+a))/(4pi in_(0)(z^(2) - a^(2))) = (q2a)/(4pi in_(0)(z^(2) - a^(2))) = (p)/(4pi in_(0) (z^(2) - a^(2)))` Potential at (x,y,0), i.e., at a point `_|_` to z-axis where charges are located, is zero. (ii)We haveprovedthat`V = (p COS theta)/(4pi in_(0) (r^(2) - a^(2) cos^(2) theta))` If `(r)/(a) gt gt 1`, then `a LT ltr :. V = (p cos theta)/(4pi in_(0) r^(2)) :.Vprop (1)/(r^(2))` i.e.,POTENTIALIS inverselyproportionalto squareof the distance. (III) Potential at `(5,0,0) is V_(1) = (-q)/(4 pi in_(0)) (1)/(sqrt((5-0)^(2) + (-a)^(2))) + (q)/(4pi in_(0)) (1)/(sqrt((5-0)^(2) + a^(2)))` `= (-q)/(4pi in_(0) sqrt(25+a^(2))) + (q)/(4pi in_(0) sqrt(25+a^(2))) = zero` Potential at `(-7,0,0) is V_(2) =(-q)/(4pi in_(0)) (1)/(sqrt((-7 - 0)^(2) + a^(2))) + (q)/(4pi in_(0)) (1)/(sqrt((-7 - 0)^(2) + a^(2))) = zero`. As work done = charge `(V_(2) - V_(1)) :.`W = zero. As work doen by electrostaicfield is independentof the path connectingthe twopoints, therefore, work doen willcontinue to be zero along every path. |
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