Consider the following charge configuration Assume that the

Consider the following charge configuration. Assume that the distance from q1 to the origin is the same as the distance from q2 to the origin. A) Find the magnitude of the resulting Electric field at the origin of the drawn coordinate system in terms of q, k, D, where k is the coulomb’s law constant. Any Sine and Cosine factors should be expressed in terms of the geometry of the problem. In other words, there should not be any sine or cosine functions in your final answer. However, it is perfectly OK (and desirable) to use them in your intermediate calculations. (Hint: Use the symmetry of the problem.) B) Find the angle that the resulting E field makes with respect to the x axis in terms of q and possibly any other of the problem parameters q, k, D. You may assume that all the charges are positive and equal to q.

Solution

We will find the resultant field first by tilting the x- y axis about origin by angle theta in clockwise direction so that q3 lies on the y axis, and then this field will be expressed in x-y axis as in the figure.

In the tilted frame:

Horizontal component of field due to q1 is cancelled by q3, and hence no horizontal component of the field in the final field
y - field due to q1 = y - field due to q3
Field at origin due to q1 and q3, E1 = - 2 * k * q *cos (phi) (j)/ ((2D)^2 + (D)^2)
where j is unit vector in y direction, phi is the angle which the line q1 - origin makes with y - axis
cos (phi) = 2 / sqroot(5), phi = 26.56 degree

E1 = - 1.788 *k *q (j) / D^2 * 5 = - 0.3576 * k * q j / D^2

E2 (due to q2) = - k * q j / (2.5D)^2 = - 0.16 k * q j / D^2

E (total) = E1 + E2 = - 0.5176 * k * q j / D^2 ( in the tilted frame of reference)

In the Normal frame, this field will have same magnitude but would be inclined at an angle theta to the -ve y axis in the clockwise direction (Because the field in the tilted frame is along -ve y axis which was at angle theta in the clockwise direction w.r.t the original -ve y axis)

Hence E total = 0.5176 * k * q / D^2 at an angle 90 + theta wrt x axis in the clockwise direction.