InterviewSolution
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InterviewSolution
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Find the potential difference `V_(AB)` between `A (0,0,0) and B (1 m , 1 m,1 m)` in an electric field : (i) `vec E = (y hat i + x hat j) Vm^-1` (ii) `vec E = (3 x^2 y hat i + x^3 hat j) Vm^-1`. |
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Answer» (a) `dV = -(E_(x) dx + E_(y) dy) = -(ydx + xdy)` `int y dx = yx` or `int xdy = xy` Since integer of both terms is same i.e. `vec(E ) . D vec(r )` is a perfect differential. `V_(1,2,3) - V_(0,0,0) = - int_(0,0,0)^(1,2,3) vec(E ) . d vec(r ) = - int(E_(x) dx + E_(y) dy)` `= - int (y dx + xdy) = - int d(xy) = -xy` `= -|xy|_(0 ,0)^(1,2)` `= - [ (1 xx 2) - (0 , 0)] = -2` `= -2` volt (b) `dV = -(E_(x) dx + E_(y) dx)` ` = -(3 x^(2) y dx + x^(3) dy)` `int3x^(2) y dx = y int 3 x^(2) dx = y . 3 (x^(3))/(3) = x^(3) y` `int x^(3) dy = x^(3) int dy = x^(3) y` Since integral of both terms is same , i.e. `vec(E ) . d vec(r)` is perfect differential. `V_(1,2,3) - V_(0,0,0) = - int_(0,0,0)^(1,2,3) (3x^(2) y dx + x^(3) dy) = - int_(0,0,0)^(1,2,3) d(x^(3) y)` `= - [x^(3) y ]_(0,0)^(1,2) = - [(1^(3) xx 2) - 0 xx 0)]` ` = - 2` volt |
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