A semicircle of radius a is in the first and second quadrants, with the center o

Question

A semicircle of radius a is in the first and second quadrants, with the center of curvature at the origin. Positive charge +Q is distributed uniformly around the left half of the semicircle, and negative charge ?Q is distributed uniformly around the right half of the semicircle in the following figure. What is the magnitude of the net electric field at the origin produced by this distribution of charge? Please express your answer in terms of variables Q, a, constant (pi), and electric constant? (subscript 0)

Explanation / Answer

There's a lot of symmetry here. The positive Q charge is centered at (a, 3?/4) in polar coordinates, and the negative charge is centered at (a, ?/4).

The field due to the +Q charge has magnitude kQ/a^2 and direction -?/4 (4th quadrant).

The field due to the -Q charge has the same magnitude, but the direction is +?/4.

When you add these components together, the direction is along the +x-axis (Answer b) because the vertical components cancel out. The magnitude of the field is kQ?2 / a^2

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