Problem 18 - A Complete Resource Book in Physics for JEE Main

Question

There are two charges \(+1 \mu \mathrm{C}\) and \(+5 \mu \mathrm{C}\). The ratio of the forces acting on them will be (A) \(1: 5\) (B) \(1: 1\) (C) \(5: 1\) (D) \(1: 25\)

Step-by-Step Solution

Answer

The ratio of the forces acting on the charges is \(1:1\), which corresponds to option (B).

Step 1: Recall Coulomb's Law

Coulomb's Law states that the force between two charged particles is given by: \(F = k \dfrac{q_1 \cdot q_2}{R^2}\) Where: - F is the force between the charged particles. - k is Coulomb's constant (\(k = 8.987 \times 10^9 Nm^2/C^2\)). - \(q_1\) and \(q_2\) are the magnitudes of the charges. - R is the distance between the charged particles.

Step 2: Write the forces acting on each charge

Let's denote F1 as the force acting on the \(+1\mu C\) charge, and F2 as the force acting on the \(+5\mu C\) charge. According to Coulomb's Law, the forces acting on these charges are proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. Therefore, \(F1 \propto (1 \mu C) \cdot (5 \mu C)\) and, \(F2 \propto (5 \mu C) \cdot (1 \mu C)\)

Step 3: Determine the ratio of forces

The proportionality allows us to write the ratio of the forces as: \(\dfrac{F1}{F2} = \dfrac{(1 \mu C) \cdot (5 \mu C)}{(5 \mu C) \cdot (1 \mu C)}\)

Step 4: Simplify the ratio and select the correct answer

Simplify the expression: \(\dfrac{F1}{F2} = \dfrac{5}{5} = 1\) So the ratio of the forces acting on the charges is \(1:1\), which corresponds to option (B).