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You are given two concentric charged conducting spheres of radii R_(1) and R_(2) such that R_(1) gt R_(2) ,having charges Q_(1) and Q_(2) respectively and uniformly distributed over its surface. Calculate E and V at three points A, B , C whose distances from the centre are r_(A), r_(B) and r_(c)respectively as shown in figure. |
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Answer» Solution :At point A , the point A is outside both the spheres so for the purpose of calculation of E and V we have to consider both `Q_(1) and Q_(2)` to be CONCENTRATED at the centre. So, E ` ((Q_(1) + Q_(2)))/(4 pi epsilon_(0).r_(A)^(2)) and V = ((Q_(1) + Q_(2)))/(4 pi epsilon_(0).r_(A))` At point B : For ELECTRIC field calculation note that the point B is inside the outer sphere so there will be no contribution of `Q_(1)` in the electric field at B. But for `Q_(2)` charges, the point B is outside so we have t assume `Q_(2)` to be at the centre and distance of B will be taken to be `r_(B)` So, `e at B = (Q_(2))/(4 pi epsilon_(0)r_(B)^(2))` Now, for potential, remember that a single charge will never give zero contribution to potential at finite pionts. So `Q_(1)` as well as `Q_(2)` will contribute to potential at B. Noe talk about the contribution of `Q_(1)` at point B you can SEE that the point B is inside the large sphere of `Q_(1) `so `Q_(1)` will OFFER its surface potential`(Q_(1))/(4 pi epsilon_(0)R_(1)` at all its inside points. but what about the potential contribution of `Q_(2)` at point B! for `Q_(2)` the point B is outside the sphere so, we'll assume as if `Q_(2)` is concentrated at the centre and the distance taken will be `r_(B)`. so the contribution of `Q_(2)` in potential will be `(Q_(2))/(4 pi epsilon_(0)r_(B))` Hence the total potential at B = `(Q_(1))/(4 pi epsilon_(0)R_(1)) + (Q_(2))/(4 pi epsilon_(0)r_(B))` At point C : For electric field note that to both `Q_(1) and Q_(2)` . the point c is inside. So, the net contribution in the electric field will be zero, i.e E at C = zero. infact, at all the point inside the smaller sphere, the electric field will be zero, whate about potential at C! For both the spheres the point C is inside so both `Q_(1) and Q_(2)` will offer their respective surface potential. since surface potential of `Q_(1)` is `(Q_(1))/(4 pi epsilon_(0)R_(1))` and that of `Q_(2)` is `(Q_(2))/(4 pi epsilon_(0)R_(2))` , hence the potential at C will be `(Q_(1))/(4 pi epsilon_(0)R_(1)) + (Q_(2))/(4 pi epsilon_(0)R_(2))` In fact at all the points inside the inner sphere the potential will be constant and will be equal to the potential at C. |
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