Problem 1: Find the force between charges q_{1} = +4 μC and q_{2} = −6 μC separated by r = 30 cm.

Step 1 — Identify given quantities with units: q_{1} = +4 μC = $4 \times 10^{-6} C q_{2} = −6 μC = \6 \times 10^{-6} C (magnitude) r = 30 cm = 0.30 m k = \9 \times 10^{9}$N$m^{2}$$C^{-2}$

Step 2 — Apply Coulomb's law (magnitude): $F = \frac{k|q_1||q_2|}{r^2} = \frac{9\times10^9\ \text{N m}^2\text{C}^{-2} \times 4\times10^{-6}\ \text{C} \times 6\times10^{-6}\ \text{C}}{(0.30\ \text{m})^2}$

Step 3 — Calculate numerator: Numerator = $9 \times 10^{9} × \4 \times 10^{-6} × \6 \times 10^{-6} = 9 × \24 \times 10^{-3} = \216 \times 10^{-3}$= 0.216 N$m^{2}$

Step 4 — Calculate denominator: $r^{2}$ = (0.30)^{2} = 0.09 $m^{2}$

Step 5 — Final result: $F = \frac{0.216\ \text{N m}^2}{0.09\ \text{m}^2} = 2.4\ \text{N (attractive, since charges are opposite)}$