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An infinite straight wire carrying linear charge density lambda1 is placed along the y-axis. A semicircular wire of radius R carrying uniform linear charge density lambda2 is centred at the origin with its diameter along the y-axis (forming a D-shape in the x-y plane opening toward positive x). Find the ratio lambda1 / lambda2 such that the net electric field at the centre of the semicircle (the origin) is zero.
- lambda1 / lambda2 = pi / 2
- lambda1 / lambda2 = pi
- lambda1 / lambda2 = 2 / pi
- lambda1 / lambda2 = 1 / pi
Correct answer: lambda1 / lambda2 = pi
Solution
The infinite wire along y-axis produces a horizontal field (in +x direction) at the point (R,0). The semicircular arc (in the first and fourth quadrants, centred at origin) produces a field pointing in the -x direction at the origin. For these to cancel, their magnitudes must be equal.
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