Problem 66 - Arihant AIEEE Physics

Question

A point charge is projected along the axis of circular ring of charge \(Q\) and radius \(10 \sqrt{2} \mathrm{~cm}\). The distance of the point charge from centre of ring, where acceleration of charged particle is maximum, will be : (a) \(10 \mathrm{~cm}\) (b) \(20 \mathrm{~cm}\) (c) at infinity (d) none of these

Step-by-Step Solution

Answer

The distance is 10 cm.

Step 1: Understand the Problem

We need to find the position along the axis of a charged ring where the acceleration of a point charge is maximum. The ring has charge \( Q \) and radius \( R = 10\sqrt{2} \text{ cm} \).

Step 2: Recall the Formula for Electric Field

The electric field \( E \) at a point located at a distance \( x \) from the center along the axis of the ring is given by: \[ E = \frac{kQx}{(x^2 + R^2)^{3/2}} \] where \( k \) is Coulomb's constant.

Step 3: Relate Electric Field to Force and Acceleration

The force \( F \) on a point charge \( q \) due to the electric field is \( F = qE \). Hence, the acceleration \( a \) of the charge is given by \( a = \frac{F}{m} = \frac{qE}{m} \), where \( m \) is the mass of the point charge.

Step 4: Determine Maximum Acceleration

The acceleration \( a = \frac{qkQx}{m(x^2 + R^2)^{3/2}} \) is maximum when the derivative of \( E \) with respect to \( x \) is zero. This involves setting \( \frac{d}{dx} \left( \frac{kQx}{(x^2 + R^2)^{3/2}} \right) = 0 \).

Step 5: Apply Derivative and Find Critical Points

Find the derivative: \( E'(x) = \left( \frac{d}{dx} \right) \left( \frac{kQx}{(x^2 + R^2)^{3/2}} \right) \). Setting \( E'(x) = 0 \) gives the condition \( R^2 = 2x^2 \), which solves to \( x = \frac{R}{\sqrt{2}} \).

Step 6: Calculate the Exact Position

Given \( R = 10\sqrt{2} \text{ cm} \), solve \( x = \frac{R}{\sqrt{2}} \) to find \( x = 10 \text{ cm} \).

Step 7: Verify Solution Against Options

The calculated position \( 10 \text{ cm} \) matches option (a). Therefore, the distance of the point charge from the center of the ring where the acceleration is maximum is \( 10 \text{ cm} \).

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electromagnetism that describes the force between two point charges. It states that the magnitude of the electric force between two stationary, electrically charged particles is directly proportional to the product of the charges. Furthermore, it is inversely proportional to the square of the distance between the centers of the two charges. The formula used to express this relationship is: \[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \(F\) is the magnitude of the force between the charges,
• \(k\) is Coulomb's constant, approximately equal to \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\),
• \(q_1\) and \(q_2\) are the amounts of the two charges,
• \(r\) is the distance between the charges.

This force can either attract or repel the charges depending on their signs. If the charges are of opposite sign, they attract each other. If they have the same sign, they repel each other. It is important to note that Coulomb's Law applies to point charges, which are idealizations of particles with charge that have no size.

Point Charge

A point charge is an idealized model of a particle with an electric charge. It is considered to have negligible size compared to the distances involved in the problem. Point charges are used to simplify complex problems in electrostatics where the actual size and shape of the charged object do not significantly affect the calculation of forces and fields.
In practical applications, we use point charges to model the behavior of charges in a vacuum where the forces are assumed to act directly at the centre of the charge. The concept of a point charge is essential in theoretical physics because it allows us to calculate electric fields and forces using simpler mathematical expressions. This simplification is crucial for understanding fundamental electric interactions without the complications introduced by extended charge distributions. When dealing with point charges, forces and electric fields become more straightforward, fitting perfectly into the framework established by Coulomb's Law.

Electric Force

Electric force is a fundamental interaction that arises between charged particles. According to Coulomb's Law, this force can either be attractive or repulsive, exerted over a distance between the charges. It is crucial to understand that electric force is a vector quantity, meaning that it has both magnitude and direction. This characteristic makes it essential to consider the orientation of charges when analyzing physical problems.
The electric force acting on a point charge \( q \) placed in an electric field \( E \) can be expressed by the equation:\[ F = qE \]Here, \( F \) represents the electric force exerted on the charge, while \( E \) is the electric field strength at the location of the charge. This formula illustrates how the field affects the charge, with the force being parallel to the direction of the field.

• If \( q \) is positive, the force will act in the same direction as the electric field.
• Conversely, if \( q \) is negative, the force will act in the opposite direction.

Understanding the electric force is fundamental in physics because it governs not only interactions between static charges but also the behavior of charges in motion, playing a significant role in understanding phenomena such as electricity and magnetism.