Question

Two identically charged particles separated by a distance of \(1.00 \mathrm{~m}\) repel each other with a force of \(1.00 \mathrm{~N}\). What is the magnitude of the charges?

Short answer

Answer: The magnitude of the charges on the particles is approximately \(1.07 \times 10^{-5} \mathrm{C}\).

Step-by-step solution

Write down Coulomb's Law formula

Coulomb's Law states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, Coulomb's Law is given by: $$F = k \frac{q_1 q_2}{r^2}$$ where \(k\) is Coulomb's constant (\(8.99 \times 10^9 \frac{\mathrm{N \cdot m^2}}{\mathrm{C^2}}\)), \(q_1\) and \(q_2\) are the charges of the particles, \(F\) is the force between them, and \(r\) is the distance between the charges.

Rearrange the formula to solve for q

Since both particles have the same charge (let's call it \(q\)), we can rewrite Coulomb's Law as: $$F = k \frac{q^2}{r^2}$$ Now, we'll rearrange the formula to solve for \(q\): $$q^2 = \frac{F r^2}{k}$$ Taking square root of both sides, $$q = \sqrt{\frac{F r^2}{k}}$$

Plug in given values and calculate q

Now we can plug in the given values into the formula to find \(q\). We're given the force, \(F = 1.00 \mathrm{~N}\), distance, \(r = 1.00 \mathrm{~m}\), and we know Coulomb's constant, \(k = 8.99 \times 10^9 \frac{\mathrm{N \cdot m^2}}{\mathrm{C^2}}\). Plugging these into the formula: $$q = \sqrt{\frac{(1.00 \mathrm{~N})(1.00 \mathrm{~m})^2}{8.99 \times 10^9 \frac{\mathrm{N \cdot m^2}}{\mathrm{C^2}}}}$$

Simplify and find the magnitude of the charge

Now we can simplify the expression and find the magnitude of the charge: $$q = \sqrt{\frac{1.00 \mathrm{~N \cdot m^2}}{8.99 \times 10^9 \frac{\mathrm{N \cdot m^2}}{\mathrm{C^2}}}}$$ $$q = \sqrt{\frac{1.00}{8.99 \times 10^9} \mathrm{C^2}}$$ $$q \approx 1.07 \times 10^{-5} \mathrm{C}$$ So, the magnitude of the charges on the particles is approximately \(1.07 \times 10^{-5} \mathrm{C}\).