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# Electric field in the center of a circle?

## Hi, I am stumped on this question and wish to get some guidance: You are helping to design an electron microscope. A new device to position the electron beam is a charged circle of conducting material, divided into two half circles separated by a thin insulator; one half can have a + charge and the other a - charge. The electron beam will go through the center of the circle, and your job is to calculate the electric field in the center of the circle as a function of the amount of positive charge on one half circle, the amount of negative charge on the other, and the radius of the circle. Use integration. I am confused where to begin. I am guessing I will have to use E = (kq) / r^2 somewhere. Any ideas? Thank you!

Use angle as the variable of integration, and decompose the electric field vector between the center and any infinitesimal segment of the circle into x and y components. This is easy, since it's just Sin(a) da and Cos(a) da (while letting r = 1 to simplify things, it doesn't matter since it's a constant). Integrating those functions, we have -Cos(a) and Sin(a), for a = 0 to pi (For each half circle), or 0 and 2, which makes sense since the x component electric field should cancel out by symmetry. The same is true on the other side, but the direction of electric field is the same (attractive + repulsive). The full equation for the electric field strength should be:

E = k ((2q1) +(2q2)) / r^2

where k is the constant, q1 is charge of the + ring, q2 is the charge on the - ring, and r is the radius of the circle.
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