What is the resultant electric force acting on a charge of 5 uc located at the point (3m,4m)?

Question

We assume the existence of a uniform electric field E= 500 i N/C without specifying which charges are the source. To this field is added that of a point charge of 2 uc located at the origin. What is the resultant electric force acting on a charge of 5 uc located at the point (3m,4m)?

The answer is (4.66i + 2.88j)x10^-3 N but I'm not sure how they got it?

Answer 1

Draw a coordinate system and locate +2μC at the origin and +5μC at (3, 4) meters. The given uniform electric field points in the direction of the positive X axis. The force due to the uniform field on the 5μC charge is directed toward the X axis and is given by 500i x 5 x 10^-6 = 2.5 x 10^-3 i N ( I left out units to simplify the equation).

The 2μC charge exerts a force on the 5μC charge given by 8.99 x 10⁹ x 2x10^-6 x 5 x 10^-6 / 5² = 3.60 x 10-3 in a direction pointing away from the line connecting the two charges. This force has an X component = to

3.60 x 10^-3 x 3/5 i = 2.16 i N (where 3/5 is the cosine of the angle made by the resultant vector). The Y component = 3.60 x 10^-3 x 4/5 = 2.88 x 10^-3 j N.

Adding the X components, we get F_x = (2.5 x 10^-3 i + 2.16 x 10³ i = 4.66 x 10^-3 i N. There is only one Y component = 2.88 x 10^-3 j N.

I used the equations F = E x Q and F = kQ1 x Q2 / d².