Four point charges ''2q, q, q, and q'' are placed at the corners of a rectangle of dimensions ''a'' and ''3a'' as shown in the figure. A fifth charge ''Q'' is placed at the center of the rectangle. Our task is to compute the '''electric field''' at the center of the rectangle, and then determine the force on ''Q''.

(a) Each of the 4 charges at the corners contributes to the '''electric field E''' at the center of the rectangle. You’ll have to add these contributions by components. First, start by finding the '''magnitudes''' of all the electric field contributions you will need to add together.

| E<sub>q</sub> | = kq/r<sup>2</sup> r =

| E<sub>2q</sub> | = 2kq/r<sup>2</sup>

(b) ''Without using your calculator'', calculate sin(, cos(''''and tan( where  is the angle defined in the diagram. (Express your answer algebraically.)

sinθ = 1/√10 cosθ = 3/√10 tanθ = 1/3

(c) Now compute the '''x- and y-components''' of the relevant contributions to the electric field at the center of the rectangle.

E<sub>x</sub> = E<sub>y</sub> =

(d) What is the '''total''' electric field '''E''' at the center of the rectangle, given the particular values ''q'' = 3 C and ''a'' = 2 cm?

| E | = 2.7 * 10<sup>7</sup> N/C

(e) Finally, what is the '''force''' on the fifth charge ''Q'' due to this electric field, given ''Q'' = 4 C? Remember that force is also a vector, and you should give both its x and y components.

F<sub>x</sub> = -102 N F<sub>y</sub> = -34 N