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Chapter 13: Problem 18

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) \(1: 25\)

Short Answer

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

Step by step solution

01

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.
02

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, \(F2 \propto (5 \mu C) \cdot (1 \mu C)\)
03

Determine the ratio of forces

The proportionality allows us to write the ratio of the forces as: \(\dfrac{F1}{F2} = \dfrac{(1 \mu C) \cdot (5 \mu C)}{(5 \mu C) \cdot (1 \mu C)}\)
04

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).

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