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Unformatted text preview: 55. Consider an infinitesimal section of the rod of length dx, a distance x from the left end, as shown in the following diagram. It contains charge dq = λ dx and is a distance r from P. The magnitude of the field it produces at P is given by
dE = 1 λ dx . 4πε 0 r 2 The x and the y components are
dEx = − 1 λ dx sin θ 4πε 0 r 2 1 λ dx cos θ , 4πε 0 r 2 and
dE y = − respectively. We use θ as the variable of integration and substitute r = R/cos θ, x = R tan θ and dx = (R/cos2 θ) dθ. The limits of integration are 0 and π/2 rad. Thus, Ex = − and Ey = − λ 4πε 0 R
π2
0 λ 4πε 0 R π2
0 sin θdθ = λ cos θ 4πε 0 R π2
0 =− λ 4πε 0 R cosθdθ = − λ sin θ 4πε 0 R π /2
0 =− λ . 4πε 0 R We notice that Ex = Ey no matter what the value of R. Thus, E makes an angle of 45° with the rod for all values of R. ...
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This note was uploaded on 01/25/2011 for the course PHYSICS 17029 taught by Professor Rebello,nobels during the Fall '10 term at Kansas State University.
 Fall '10
 Rebello,NobelS
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