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Find the electric field intensity at a p...

Find the electric field intensity at a point P which is at a distance R (point lying on the perpendicular drawn to the wire at one of its end) from a semi-infinite uniformly charged wire. (Linear charge density = `lambda`).

Text Solution

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Field at point P is due to an elemental charge.
`dq(=lambdadl)`
`dE=k(dq)/((R^(2)+l^(2)))`
The component along x-axis
`dE_(x)=(kdqcostheta)/((R^(2)+l^(2)))=(k(dq)R)/((R^(2)+l^(2))^(3//2))`
`therefore E_(x)=underset(0)overset(oo)int(1)/(4piepsilon_(0))(lambdadlR)/((R^(2)+l^(2))^(3//2))impliesE_(x)=(lambda)/(4piepsilon_(0)R)`
Similarly, `E_(y)=underset(0)overset(oo)int(1)/(4piepsilon_(0))(lambda(dl)l)/((R^(2)+l^(2))^(3//2))`
`therefore E_(y)=(lambda)/(4piepsilon_(0)R)`
`therefore E=sqrt(E_(X)^(2)+E_(Y)^(2))=(lambda)/(2.sqrt(2)piepsilon_(0)R),tantheta=(E_(y))/(E_(x))`
`implies theta=45^(@)`
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