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Calculate electric field intensity at point P due to a thin positive uniformly charged rod . Point P is at perpendicular distance d from the rod . The lower and upper ends of the rod make the angles alpha and beta as shown below . |
Answer» Solution : Refer the given figure . Let us SELECT one segment of length dy at a distance y from O. Electric field due to this segment is `dvecE` as shown. We have `lambda` as linear CHARGE DENSITY .So the charge on segment is `lambdady` . `dE=1/(4piepsilon_0)(lambdady)/(d^2+y^2)` The direction of `dvecE` is dissimilar for different locations of the segment . Hence `dvecE` cannot be integrated directly. So, we shall integrate components perpendicular and parallel to rod separately . `E_x=int dE cos theta =lambda/(4piepsilon_0)int (dy)/(d^2+y^2)cos theta` `E_y=int dE sin theta =lambda/(4piepsilon_0) int (dy)/(d^2+y^2)sin theta` To solve above integrations, let us assume : `y=d tan theta` `dy=d sec^2 theta d theta ` `d^2+y^2 =d^2 sec^2 theta` `(dy)/(d^2+y^2)=(d theta)/d` On substituting , `E_x=int dE cos theta - lambda/(4piepsilon_0d) int_(-ALPHA)^(beta)cos theta d 0` `E_y=int dE sin theta =lambda/(4pi epsilon_0 d)int_(-alpha)^beta sin theta d theta` In the above integration , the limits are taken from `theta=-alpha` corresponding to lower end of rod . And `theta=beta` for the upper end of the rod . On solving both, we GET the following :`E_x = int dE cos theta = lambda/(4piepsilon_0d) int_(-alpha)^beta cos theta d theta =lambda/(4piepsilon_0d) [sin theta]_(-alpha)^(beta)` `=lambda/(4piepsilon_0d) [sin beta - sin (-alpha)]` `E_y=int dE sin theta =lambda/(4pi epsilon_0d) int_(-alpha)^beta sin theta d theta =lambda/(4pi epsilon_0 d) [-cos theta ]_(-alpha)^(beta)` `=lambda/(4piepsilon_0d)[(-cos beta)-(-cos (-alpha))]` `E_x=lambda/(4pi epsilon_0d)(sin alpha + sin beta) E_x` represents the component, perpendicular to rod in the direction away from the rod for `lambda > 0 . E_y=lambda/(4piepsilon_0d)(cos alpha -cos beta)` `E_y` represents component that is parallel to the rod . The direction of `E_y` can be upwards or downwards depending on the values of `alpha` and `beta`. |
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