Question

A point charge of +5.0 micro coulombs is located on the x-axis at x=-3.0 cm, and a second point charge of -8.0 micro coulombs is located on the x-axis at x=+4.0 cm. Where should a third charge of +6.0 Micro coulombs be placed so the the the electric field at the origin is zero?

Answer 1

here is the situation described by your problem:

so starting from this vector equation:
\[\Large {{\mathbf{E}}_1}\left( 0 \right) + {{\mathbf{E}}_2}\left( 0 \right) + {{\mathbf{E}}_3}\left( 0 \right) = {\mathbf{0}}\]
where all fields are computed at x=0, we get this condition:
\[\Large \frac{1}{{4\pi {\varepsilon _0}}}\frac{{{q_1}}}{{x_1^2}} + \frac{1}{{4\pi {\varepsilon _0}}}\frac{{{q_2}}}{{x_2^2}} + \frac{1}{{4\pi {\varepsilon _0}}}\frac{{{q_3}}}{{{d^2}}} = 0\]
where \(\large q_1, q_2, q_3 \) are your charges with their corresponding signs