Question

A semicircle of radius r 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. What is the magnitude of the net electric field at the origin produced by this distribution of charge? Express your answer in terms of the variables Q, r, and appropriate constants. Thanks in advance for any help in approaching this. Can't quite nab the hang of this one xD

Answer 1

One thing you can do is find the electric field from each quarter of a circle, and add them together. But how would you do that?

\[\vec{E} = \frac{1}{4\pi \epsilon_0} \int\limits_C \lambda_q \frac{\hat{r}}{\left|| \vec{r} \right||^2}ds = \frac{1}{4\pi \epsilon_0} \int\limits_C \lambda_q \frac{\vec{r}}{\left|| \vec{r} \right||^3}ds\]

\[\vec{r} = x\hat{x} + y\hat{y}\]

The electric field is going to have two components that you need to consider. Notice although you are integrating a vector, this is not a line integral of a vector field which produces a scalar - here we are integrating a vector to produce another vector. You do this by integrating component separately like it was a scalar field. So you will have two line integrals for each quarter of a circle. Essentially this is like getting a small little electric field vector for each point on the curve and adding them all up.

Answer 2

actually there should be a minus sign on that integral, since the vector from the charge on the curve to the origin is \[- \vec{r}\]