Answer:
a)
b)
Step-by-step explanation.
In order to solve this problem, we mus start by plotting the given points and charges. That will help us visualize the problem better and determine the direction of the forces (see attached picture).
Once we drew the points, we can start calculating the forces:
which yields:
(I will assume the positions are in meters)
Next, we can make use of the force formula:
so we substitute the values:
which yields:
Now we can find its components:
And we can now write them together for the first force, so we get:
We continue with the next force. The procedure is the same so we get:
which yields:
Next, we can make use of the force formula:
which yields:
Now we can find its components:
And we can now write them together for the second, so we get:
We continue with the next force. The procedure is the same so we get:
which yields:
Next, we can make use of the force formula:
which yields:
Now we can find its components:
And we can now write them together for the third force, so we get:
So in order to find the resultant force, we need to add the forces together:
so we get:
So when adding the problem together we get that:
which is the answer to part a), now let's take a look at part b).
b)
Basically, we need to find the magnitude of the force and divide it into the test charge, so we get:
which yields:
and now we take the formula for the electric field which is:
so we go ahead and substitute:
