Question

Work out what the combined electric field is for a line of charge and a semicircle of a charge. Positive electric charge q1 is distributed uniformly in a semicircle of radius a. Have this semicircle to go from 0 degrees to 180 degrees standard angle. Negative charge of magnitude Q2 is uniformly distributed along a straight line of length 2a. This line is horizontal and parallel to the x axis. The line of charge is at a distance y greater than a. The y axis bisects the line of charge. Determine the expression for the combined electric field at the origin which is the center of the semicircle

Answer 1

Semi circle: positive charge. Q1

Line: negative charge. Q2

Line is at a distance y> a. So it is above the semi circle. Assume the semicircle touches the x axis at the ends.

Assume that the entire arrangement lies on a plane.

The electric field by the semi circle will be pointed towards the negative y axis. This is because of symmetry. The x component of the field created one half of the semicircle will be cancelled by the x component of the other half.

let theta be the angle measured from the x axis. Thus:

where

We can integrate this from theta 0 to 180 to get the total field.

But we are interested only in the component of E along the y axis. This is

So the final expression is:

Given that the radius is a the field by the semicircle is:

The field by a finite wire is simply given by a formula:

where now

This field will point upwards since the charge is negative.

The sum total of the two electric fields is: