Question

Three electric charges are arranged on an x–y coordinate system, as shown above. Express all algebraic answers to the following parts in terms of Q, q, x, d, and fundamental constants.

a. On the diagram, draw vectors representing the forces F₁ and F₂

b. Determine the magnitude and direction of the total electric force on the +q charge. exerted on the +q charge by the +Q and –Q charges, respectively.

c. Determine the electric field (magnitude and direction) at the position of the +q charge due to the other two charges.

d. Calculate the electric potential at the position of the +q charge due to the other two charges.

e. Charge +q is now moved along the positive x–axis to a very large distance from the other two charges. The magnitude of the force on the +q charge at this large distance now varies as 1/x³. Explain why this happens.

Answer 1

a.

b. The x components of the forces cancel so the net force is the sum of the y components, which are equal in magnitude and direction. F_net = 2 × F cos θ where θ is the angle between the y axis and the dashed line in the diagram above cos θ = d/r = 

This gives F_net = 2 × kqQ/r² × cos θ = 2kqQd/(x² + d²)^3/2

c. E = F/q at the point where q₁ lies E = 2kQd/(x² + d²)^3/2

d. Since the charges Q and –Q are equidistant from the point and V = ΣkQ/r, the potential V = 0 

e. As x gets large, the distance to the charges r and the value of x become similar, that is   Substituting this into the answer to b. yields F = 2kqQd/x³