Step 1: Analyze Forces on Charge at Origin:
Let the charge at origin be \( q_0 = 2 \, \mu\text{C} \).
The forces acting on \( q_0 \) are due to \( Q \) at \( x_1=1 \), \( q_2=4 \, \mu\text{C} \) at \( x_2=2 \), and \( q_3=12 \, \mu\text{C} \) at \( x_3=4 \).
According to Coulomb's Law, force \( F = \frac{k q_1 q_2}{r^2} \).
Since \( q_0 \) is positive, repulsive forces (from positive charges) will be in the negative x-direction. For net force to be zero, attractive forces must balance repulsive ones.
Step 2: Equation for Net Force:
Sum of forces \( = 0 \).
\( \frac{k q_0 Q}{(1)^2} + \frac{k q_0 (4)}{(2)^2} + \frac{k q_0 (12)}{(4)^2} = 0 \)
(Distances in cm, but units cancel out if consistent).
Cancel \( k q_0 \) from all terms:
\[ \frac{Q}{1} + \frac{4}{4} + \frac{12}{16} = 0 \]
Step 3: Solve for Q:
\[ Q + 1 + \frac{3}{4} = 0 \]
\[ Q + 1 + 0.75 = 0 \]
\[ Q + 1.75 = 0 \]
\[ Q = -1.75 \, \mu\text{C} \]