Problem 139 - Master Resource Book in JEE Main Physics

Question

The question contains Statement I and Statement II. Of the four choices given after the statements, choose the one that best describes the two statements. Statement I For a charged particle moving from point \(P\) to point \(Q\), the net work done by an electrostatic field on the particle is independent of the path connecting point \(P\) to point \(Q\). Statement II The net work done by a conservative force on an object moving along a closed loop is zero. [AIEEE 2009] (a) Statement 1 is true, Statement 11 is false (b) Statement 1 is true, Statement II is true, Statement II is the correct explanation of Statement 1 (c) Statement 1 is true, Statement 11 is true, Statement 11 is not the correct explanation of Statement 1 (d) Statement 1 is false, Statement \(\|\) is true

Step-by-Step Solution

Answer

(c) Statement I is true, Statement II is true, Statement II is not the correct explanation of Statement I.

Step 1: Understand Statement I

Statement I claims that the net work done by an electrostatic field on a charged particle moving from point \( P \) to point \( Q \) is independent of the path taken. This is characteristic of a conservative field. In a conservative field like an electrostatic field, the work done depends only on the initial and final positions, not on the path followed.

Step 2: Analyze Statement II

Statement II asserts that the net work done by a conservative force on an object moving along a closed loop is zero. By definition, conservative forces have this property: the work done by a conservative force in a closed loop is always zero.

Step 3: Evaluate the Truth of Statements

Both Statement I and Statement II are true. Statement I describes a property of the electrostatic (conservative) field where work done is path-independent. Statement II describes the characteristic of conservative forces (including the electrostatic force) where the work done over a closed loop is zero.

Step 4: Determine the Relationship between the Statements

While both statements are true, Statement II directly supports the general characteristics of conservative forces rather than explaining why an electrostatic field in particular is path-independent. Therefore, Statement II is not the correct explanation of Statement I.

Key Concepts

Conservative Force

A conservative force is a type of force where the work done moving a particle between two points is independent of the path taken. This concept is fundamental in physics, especially when discussing forces like gravity and electrostatics. The defining feature of a conservative force is that it conserves mechanical energy. When a force is conservative, any energy used in moving an object through the force field can be fully recovered by moving the object back to its original position.

Some key characteristics of conservative forces include:

• The work done by the force depends only on the initial and final positions, not on the particular trajectory.
• The total mechanical energy (sum of kinetic and potential energy) of a system remains constant if only conservative forces do work.
• If a charged particle or mass completes a loop, the total work done by a conservative force is zero.

In electrostatics, the electric field is a classic example of a conservative force field.

Path Independence in Physics

Path independence is a crucial concept associated with conservative forces, and it means the total work done by such a force on a particle between two points is always the same, regardless of the path taken. For example, in an electrostatic field, a charged particle can move from point \(P\) to point \(Q\) through any route, and the work done will remain constant.

This property is enabled by the conservative nature of forces like the electrostatic force.
Because these forces are path-independent, predicting the work done becomes significantly simpler, as it only requires knowledge of the start and end points rather than the specific details of the path.

Path independence means:

• The work done by the force on any closed path or loop is zero.
• Energy calculations simplify since they do not depend on specific trajectories.

This fundamental principle supports many simpler calculations and allows for effective energy conservation analysis in physics problems.

Work-Energy Principle in Electrostatics

The work-energy principle connects the work done by forces to changes in kinetic energy. In the realm of electrostatics, this principle helps us understand how electrostatic forces alter the energy of charged particles. When a charged particle moves in an electrostatic field, work is done on it, resulting in changes to its potential and kinetic energies.

Here's what happens in electrostatics based on the work-energy principle:

• The work done by the electrostatic force results in a change in the particle's kinetic energy.
• The amount of work is equal to the negative change in potential energy as the particle moves.
• Conservative characteristics ensure energy transformations within the field are efficient and path-independent.

Electrostatic forces being conservative means that any change in energy due to movement between two points can be completely accounted for simply by looking at the energy state at the beginning and end.