ElectrostaticsMedium
Question
A linear positive charge distribution, with linear charge density λ coulomb per meter, extends along +x-axis from x = 0 to x = ∞.
The electric field E at any point (0, y) on the y-axis
Options
A.is proportional to λ/y2 irrespective of whether y is positive or negative.
B.is always directed away and perpendicular to the line of charge.
C.has a vanishing component parallel to the line of charge.
D.is directed along a straight line of slope m = −1 if y is positive but along a line of slope m = +1 if y is negative.
Solution
[Key Concept: At a point level with the end of a semi-infinite line charge, the parallel and perpendicular field components are equal, each λ/(4πε0|y|).]
Step 1: An element λ dx at (x, 0) gives at (0, y) a field along (−x, y)/(x2 + y2)3/2.
Step 2: Integrating x from 0 to ∞: Ex = −λ/(4πε0|y|) and Ey = λ/(4πε0y).
Step 3: For y > 0 the field points along (−1, 1), slope −1; for y < 0 along (−1, −1), slope +1. The magnitude varies as 1/|y| and the parallel component is not zero, so only Option D is right.
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