Problem 42 - Arihant AIEEE Physics

Question

Two points charges \(+q_{1}\) and \(+q_{2}\) are placed at a certain distance apart, then: (a) they produce the same electric field on each other (b) they exert same forces on each other (c) for minimum force between them, the magnitude of each charge must be equal to \(1.6 \times 10^{-19} \mathrm{C}\) (d) all of the above

Step-by-Step Solution

Answer

The correct answer is (b).

Step 1: Understand the Context

We are given two positive point charges, \(+q_1\) and \(+q_2\), placed at a certain distance apart. The options suggest examining the electric field and forces between these charges.

Step 2: Analyze Electric Fields

For option (a), consider the electric field. The electric field produced by a point charge \(q\) at a distance \(r\) is given by \(E = \frac{kq}{r^2}\) where \(k\) is Coulomb's constant. While \(+q_1\) and \(+q_2\) produce their respective electric fields, they do not produce the same field on each other unless \(q_1 = q_2\).

Step 3: Evaluate the Forces Between Charges

For option (b), use Coulomb's Law: \(F = \frac{k |q_1 q_2|}{r^2}\). The force exerted by \(q_1\) on \(q_2\) and vice versa are equal in magnitude and opposite in direction according to Newton's third law, regardless of their values, thus option (b) is true.

Step 4: Consider Conditions for Minimum Force

For option (c), according to Coulomb's Law, the force \(F\) decreases with decreasing values of the charges \(|q_1|\) and \(|q_2|\) or increasing distance \(r\). The statement about specific charge values \(1.6 \times 10^{-19} \mathrm{C}\) doesn't inherently minimize the force without context around distance \(r\).

Step 5: Conclusion on the Correct Option

Option (b) is the only true statement here: the forces are always equal and opposite as per Newton's third law. Both other options involve conditions not always applicable.

Key Concepts

The Concept of Electric Field

An electric field is a region around a charged object where a force would be exerted on other charged objects. For a point charge, the electric field is given by the formula: \[ E = \frac{kq}{r^2} \] where:

• \( E \) is the electric field,
• \( k \) is Coulomb's constant (approximately \( 8.988 \times 10^9 \, \text{Nm}^2/\text{C}^2 \)),
• \( q \) is the charge producing the field,
• \( r \) is the distance from the charge.

It's crucial to understand that every charge creates its own electric field, regardless of other charges nearby. Thus, two different charges, plus on a generally equal level of terms, will not produce identical electric fields on each other except when they are equal in magnitude.

Newton's Third Law and its Application

Newton's third law is a fundamental principle that states: "For every action, there is an equal and opposite reaction." In the context of electrostatics, when a charge \( q_1 \) exerts a force \( F \) on charge \( q_2 \), \( q_2 \) simultaneously exerts an equal and opposite force \(-F \) on \( q_1 \).
This means:

• The magnitudes of these forces are identical.
• The directions are opposite.

Coulomb's Law gives us the magnitudes of these forces:\[ F = \frac{k |q_1 q_2|}{r^2} \]This ensures that for any two charged particles, the force one exerts on the other is always matched by an equal and opposite force, highlighting the profound nature of Newton's third law in electrostatics.

Understanding Point Charges

Point charges are idealized charges that are assumed to be concentrated at a single point in space. This simplification allows us to more easily calculate electric fields and forces, as they disregard the complexities of the spatial extent of physical objects.
For two point charges like \( +q_1 \) and \( +q_2 \):

• They exist at specific locations.
• They generate electric fields in all directions.
• They interact with each other through these fields.

The assumption of point charges makes it straightforward to apply formulas such as Coulomb's Law and the formula for electric fields without needing to consider charge distribution.

Forces Between Charges and Coulomb's Law

Forces between point charges are governed by Coulomb's Law, which is fundamental in understanding electrical interactions. Coulomb's Law is expressed as:\[ F = \frac{k |q_1 q_2|}{r^2} \]where:

• \( F \) is the force between the charges,
• \( k \) is Coulomb's constant,
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
• \( r \) is the separation distance between the charges.

Factors that affect the force include:

• The magnitude of each charge — larger charges result in larger forces.
• The distance between charges — closer charges exert stronger forces.
• The sign of the charges — like charges repel, unlike charges attract.

Coulomb's Law intricately connects these aspects, making it essential in calculating forces in an electrostatic situation.