The question contains Statement I and Statement II. Of the four choices given after the statements,

Q: Why is the electrostatic force considered a conservative force?

The electrostatic force is conservative because it is a central force that depends only on the distance between charges. Mathematically, the curl of a static electric field is zero ($\nabla \times \vec{E} = 0$), which guarantees that the work done is path-independent.

Q: What is an example of a non-conservative force?

Friction is a classic non-conservative force. If you slide a block from point A to point B, the work done by friction depends heavily on the length of the path. A longer, winding path will result in more work done by friction than a straight path.

Q: Why isn't Statement II the correct explanation? It seems related!

While related, Statement II only states a property of conservative forces. It does not explicitly state that the electrostatic force *is* a conservative force. Without that crucial premise, it cannot logically explain why the electrostatic field exhibits path independence.

Q: Does work done being zero in a closed loop always mean the force is conservative?

Yes! In classical mechanics, a force is conservative if and only if the total work done by that force on a particle moving around any closed loop is exactly zero.

Q: What if the electric field is induced by a changing magnetic field? Is it still conservative?

Excellent question! No, an induced electric field (due to a changing magnetic field) is non-conservative. The work done in a closed loop in such a field is not zero (it equals the induced EMF). However, the problem specifically mentions an 'electrostatic' field, which implies static charges and a conservative nature.

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.

Options:

(A) Statement I is true, Statement II is false

(B) Statement I is true, Statement II is true; Statement II is not the correct explanation of Statement I

(C) Statement I is true, Statement II is true; Statement II is the correct explanation of Statement I

(D) Statement I is false, Statement II is true

Sigma Percentile · Accepted Answer

### The Enigma of Conservative Forces

When we dive into the world of classical mechanics and electrostatics, one of the most elegant and profound concepts we encounter is the idea of a **conservative force**. But what exactly makes a force "conservative"?

Imagine you are hiking up a mountain. You can take the steep, direct trail, or you can take the long, winding scenic route. Regardless of which path you choose, once you reach the summit, your change in gravitational potential energy is exactly the same. The work done against gravity depends solely on your initial and final altitudes, not on the path you took. This is the hallmark of a conservative force.

### Path Independence: The Core of Statement I

Statement I of our problem brings this concept into the realm of electrostatics. It 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 connecting them.

Is this true? Absolutely. The electrostatic force is a central force, meaning it acts along the line joining two charges and its magnitude depends only on the distance between them. Because of this, the work done by the electrostatic field can be expressed as the negative change in electrostatic potential energy:

$$W_{P 	o Q} = -(U_Q - U_P) = q(V_P - V_Q)$$

Notice how the equation only cares about the potential at $P$ ($V_P$) and the potential at $Q$ ($V_Q$). The actual trajectory—whether it's a straight line, a zigzag, or a spiral—completely vanishes from the math. Therefore, Statement I is a universally true fact for static electric fields.

### The Closed Loop: Unpacking Statement II

Statement II shifts our focus to a mathematical consequence of conservative forces. It states that the net work done by a conservative force on an object moving along a closed loop is zero.

Let's think about this logically. If the work done depends only on the initial and final positions, what happens if the initial and final positions are the exact same point?

$$W_{	ext{closed loop}} = U_{	ext{initial}} - U_{	ext{final}} = U_P - U_P = 0$$

Mathematically, this is written as a closed contour integral:

$$\oint \vec{F}_c \cdot d\vec{r} = 0$$

If you push a charge around a closed circuit in an electrostatic field, the field might do positive work on the way out, but it will do an equal amount of negative work on the way back. The net energy exchange is zero. Thus, Statement II is also a universally true definition of conservative forces.

### The Logical Trap: Assertion vs Reason

Now we arrive at the crux of the problem. We have established that both Statement I and Statement II are true. But is Statement II the *correct explanation* for Statement I?

This is where many students fall into a logical trap. Let's look at the structure of the argument:

**Fact A (Statement I):** Work done by an electrostatic field is path-independent.
**Fact B (Statement II):** Work done by a conservative force in a closed loop is zero.

Does Fact B explain Fact A? Not quite. Fact B is simply stating a property of conservative forces. It never explicitly establishes that the electrostatic field *is* a conservative force.

For Statement II to be the perfect explanation, it needed to bridge the gap. It should have said, "The electrostatic force is a conservative force, and therefore its work done is path-independent." Because Statement II merely states a generic property of conservative forces without linking it to the electrostatic field, it fails to act as the logical reason for Statement I.

Therefore, while both statements are true, Statement II is not the correct explanation of Statement I. This makes option (b) the correct choice. Always remember to look for the "missing link" in assertion-reason questions!

Select Answer:

Visualized Solution

Analyzing Statement I

Path Independence of Work

Analyzing Statement II

Connecting the Statements

The Missing Link

The Sigma Insight: Electrostatic Potential Energy

The Enigma of Conservative Forces

Path Independence

The Closed Loop

The Logical Trap

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