Find Electric Field Magnitude at Center of Charged Semi-Circle

[cp255]

Homework Statement

Positive charge Q is uniformly distributed around a semicircle of radius a.

Find the magnitude of the electric field at the center of curvature P.
Express your answer in terms of the given quantities and appropriate constants.

Homework Equations

The Attempt at a Solution

So here is my thinking. The charge per unit length of the circle would be Q/(a*pi) because a*pi is the arc length of the semi-circle. Next I assumed that the x-component is zero due to symmetry so I only need to calculate the force in the y direction. Therefore the magnitude of the electric field created by each point on the circle at the origin would be kQ/(a^2 * pi * a). The y-component of that electric field would be sin(theta)kQ/(a^3 * pi). I integrated this from theta = 0 to pi and the answer was 2kQ/(a^3 * pi). Where did I go wrong?

 

[Curious3141]

Homework Statement

Positive charge Q is uniformly distributed around a semicircle of radius a.

Find the magnitude of the electric field at the center of curvature P.
Express your answer in terms of the given quantities and appropriate constants.

Homework Equations

The Attempt at a Solution

So here is my thinking. The charge per unit length of the circle would be Q/(a*pi) because a*pi is the arc length of the semi-circle. Next I assumed that the x-component is zero due to symmetry so I only need to calculate the force in the y direction.

Right up to this point. Except I would replace "force" with "component of electric field strength". Field strength is not the same thing as force.

Therefore the magnitude of the electric field created by each point on the circle at the origin would be kQ/(a^2 * pi * a). The y-component of that electric field would be sin(theta)kQ/(a^3 * pi). I integrated this from theta = 0 to pi and the answer was 2kQ/(a^3 * pi). Where did I go wrong?

This part is wrong. "Q" here represents total charge of the semicircle. You need to find a way to represent the charge of a small element along the ring, and then work out the electric field due to it at P.

Represent the charge density per unit length by ##\sigma##. Now what is the electric field strength exerted by a charged element of length ##dl## at point P? What is the y-component? Using radian measure, how is the arc length ##l## related to ##a## and ##\theta##? Before integrating you need to make sure the only variable is ##\theta##.