1. A charge is moved from point \(A\) to point \(B\) in the electrostatic field of fixed source charges. The charge may be moved along several different paths. Which statement best describes the work done by the electrostatic force?
ⓐ. It depends only on the initial and final positions of the charge
ⓑ. It depends mainly on the length of the path followed by the charge
ⓒ. It depends on the time taken to move the charge between the points
ⓓ. It depends on the number of bends present in the chosen path
Correct Answer: It depends only on the initial and final positions of the charge
Explanation: In an electrostatic field produced by fixed charges, the electrostatic force is conservative. For a conservative force, work done between two points is path independent. This means that a long curved path and a short straight path between the same two points give the same work by the electrostatic force. The work depends on the positions \(A\) and \(B\), not on the detailed route between them. This property allows electrostatic potential energy to be defined consistently. If the work depended on the path, a single-valued potential energy at each point could not be assigned. Therefore, only the initial and final positions decide the electrostatic work.
2. A test charge is taken from point \(P\) to point \(Q\) along path \(1\), and then from \(Q\) back to \(P\) along a different path \(2\) in an electrostatic field. What is the total work done by the electrostatic force in the complete round trip?
ⓐ. Positive if path \(1\) is shorter than path \(2\)
ⓑ. Negative if path \(2\) is longer than path \(1\)
ⓒ. Equal to the work done along path \(1\) alone
ⓓ. Zero for the complete closed path
Correct Answer: Zero for the complete closed path
Explanation: \(\textbf{Given situation:}\) The charge starts from \(P\), reaches \(Q\), and finally returns to \(P\).
\(\textbf{Key principle:}\) Electrostatic force is conservative, so work done depends only on initial and final positions.
\(\textbf{Closed path condition:}\) In the complete trip, the initial point and final point are both \(P\).
\(\textbf{Work over a closed path:}\) For a conservative force, the total work over any closed path is \(0\).
\(\textbf{Symbolic statement:}\) \(\oint \vec{F}_{\text{electrostatic}}\cdot d\vec{l}=0\)
\(\textbf{Meaning:}\) The work from \(P\) to \(Q\) is exactly cancelled by the work from \(Q\) back to \(P\), even if the return path is different.
\(\textbf{Final result:}\) The total work done by the electrostatic force in the round trip is \(0\).
3. Three paths connect the same points \(A\) and \(B\) in an electrostatic field. The work done by the electrostatic force along path \(1\) is \(12\,\text{J}\). What is the work done by the electrostatic force along path \(2\) and path \(3\)?
ⓐ. \(6\,\text{J}\) and \(18\,\text{J}\)
ⓑ. \(12\,\text{J}\) and \(12\,\text{J}\)
ⓒ. \(0\,\text{J}\) and \(12\,\text{J}\)
ⓓ. \(18\,\text{J}\) and \(6\,\text{J}\)
Correct Answer: \(12\,\text{J}\) and \(12\,\text{J}\)
Explanation: \(\textbf{Known fact:}\) The electrostatic work from \(A\) to \(B\) along one path is \(12\,\text{J}\).
\(\textbf{Principle used:}\) Work done by electrostatic force is independent of path.
\(\textbf{Reason:}\) The electrostatic force due to fixed charges is a conservative force.
\(\textbf{Applying path independence:}\) If the starting point is \(A\) and the ending point is \(B\), all paths must give the same electrostatic work.
\(\textbf{Calculation:}\) \(W_{A\to B,\text{ path }2}=W_{A\to B,\text{ path }1}=12\,\text{J}\).
\(\textbf{Calculation:}\) \(W_{A\to B,\text{ path }3}=W_{A\to B,\text{ path }1}=12\,\text{J}\).
\(\textbf{Final result:}\) The work done along path \(2\) and path \(3\) is \(12\,\text{J}\) and \(12\,\text{J}\).
4. Which statement is the most accurate consequence of electrostatic force being conservative?
ⓐ. A charge cannot gain kinetic energy in an electrostatic field
ⓑ. The force on a charge must always remain constant in magnitude
ⓒ. The work done over a closed path can have any non-zero value
ⓓ. A potential energy function can be associated with the field
Correct Answer: A potential energy function can be associated with the field
Explanation: A conservative force allows the definition of a potential energy function. This is possible because the work done by the force between two points is independent of the path followed. For electrostatic force, the change in potential energy depends only on the initial and final positions of the charge. The force may vary from point to point, so conservative force does not mean constant force. A charge can also gain or lose kinetic energy depending on how electrostatic potential energy changes. The special feature is not constant motion, but path-independent work. Hence electrostatic interactions can be described using potential energy.
5. A charge moves around a closed rectangular path in the electrostatic field of fixed charges. If the electrostatic work along three sides is \(5\,\text{J}\), \(-2\,\text{J}\), and \(7\,\text{J}\), what must be the electrostatic work along the fourth side?
ⓐ. \(-10\,\text{J}\)
ⓑ. \(10\,\text{J}\)
ⓒ. \(-14\,\text{J}\)
ⓓ. \(14\,\text{J}\)
Correct Answer: \(-10\,\text{J}\)
Explanation: \(\textbf{Given work values:}\) Three parts of the closed path have work \(5\,\text{J}\), \(-2\,\text{J}\), and \(7\,\text{J}\).
\(\textbf{Unknown work:}\) Let the work on the fourth side be \(W_4\).
\(\textbf{Closed path rule:}\) For electrostatic force, total work over a closed path is \(0\).
\(\textbf{Equation:}\) \(5+(-2)+7+W_4=0\)
\(\textbf{Simplification:}\) \(5-2+7=10\), so \(10+W_4=0\).
\(\textbf{Solving:}\) \(W_4=-10\,\text{J}\).
\(\textbf{Unit check:}\) Work is measured in \(\text{J}\), and the negative sign shows that the fourth part cancels the positive net work of the other three parts.
\(\textbf{Final result:}\) The electrostatic work along the fourth side is \(-10\,\text{J}\).
6. Two different paths connect points \(A\) and \(B\). Along path \(X\), the work done by the electrostatic force is \(W_X\). Along path \(Y\), the work done by the electrostatic force is \(W_Y\). Which relation must hold?
ⓐ. \(W_X>W_Y\) if path \(X\) is longer
ⓑ. \(W_X<W_Y\) if path \(Y\) is curved
ⓒ. \(W_X=W_Y\) for the same endpoints
ⓓ. \(W_X=-W_Y\) for the same endpoints
Correct Answer: \(W_X=W_Y\) for the same endpoints
Explanation: \(\textbf{Physical setting:}\) Both paths begin at \(A\) and end at \(B\).
\(\textbf{Useful principle:}\) Electrostatic work is path independent because electrostatic force is conservative.
\(\textbf{Meaning of path independence:}\) The work is controlled by the endpoints, not by path length, shape, or curvature.
\(\textbf{Applying to the two paths:}\) Since both paths have the same initial and final positions, they must give the same work.
\(\textbf{Relation obtained:}\) \(W_X=W_Y\)
\(\textbf{Important caution:}\) The relation \(W_X=-W_Y\) would apply only to motion between opposite endpoints, such as \(A\to B\) and \(B\to A\), not to two paths with the same direction.
\(\textbf{Final result:}\) \(W_X=W_Y\) for the same endpoints.
7. A positive charge is moved slowly in an electrostatic field by an external agent. If the electrostatic force does \(+8\,\text{J}\) of work during the displacement, what is the corresponding change in electrostatic potential energy?
ⓐ. \(+8\,\text{J}\)
ⓑ. \(0\,\text{J}\)
ⓒ. \(-8\,\text{J}\)
ⓓ. \(+16\,\text{J}\)
Correct Answer: \(-8\,\text{J}\)
Explanation: \(\textbf{Known value:}\) Work done by the electrostatic force is \(W_{\text{field}}=+8\,\text{J}\).
\(\textbf{Energy relation:}\) For a conservative force, the change in potential energy is the negative of work done by the force.
\(\textbf{Formula:}\) \(\Delta U=-W_{\text{field}}\)
\(\textbf{Substitution:}\) \(\Delta U=-(+8\,\text{J})\).
\(\textbf{Simplification:}\) \(\Delta U=-8\,\text{J}\).
\(\textbf{Interpretation:}\) When the electrostatic force itself does positive work, electrostatic potential energy decreases.
\(\textbf{Sign check:}\) The negative sign belongs to the change in energy, not to the unit.
\(\textbf{Final result:}\) The change in electrostatic potential energy is \(-8\,\text{J}\).
8. A charge is moved from \(A\) to \(B\) in an electrostatic field. The work done by the electrostatic force for this movement is \(W\). What is the work done by the electrostatic force if the same charge is moved from \(B\) back to \(A\) along any path?
ⓐ. \(W\)
ⓑ. \(0\)
ⓒ. \(2W\)
ⓓ. \(-W\)
Correct Answer: \(-W\)
Explanation: \(\textbf{Given:}\) Work by electrostatic force from \(A\) to \(B\) is \(W\).
\(\textbf{Conservative-force rule:}\) The total electrostatic work over a closed path must be \(0\).
\(\textbf{Closed trip idea:}\) A movement \(A\to B\) followed by \(B\to A\) forms a closed path.
\(\textbf{Let return work be:}\) \(W_{B\to A}\).
\(\textbf{Equation:}\) \(W+W_{B\to A}=0\)
\(\textbf{Solving:}\) \(W_{B\to A}=-W\).
\(\textbf{Path note:}\) The return path may be different from the first path, but the endpoint reversal fixes the sign.
\(\textbf{Final result:}\) The work done during \(B\to A\) is \(-W\).
9. Which pair of statements correctly describes work done by electrostatic force and a typical non-conservative force such as friction?
ⓐ. Electrostatic work is path independent; frictional work generally depends on path length
ⓑ. Electrostatic work depends on path length; frictional work depends only on endpoints
ⓒ. Both electrostatic work and frictional work are always zero on a closed path
ⓓ. Both electrostatic work and frictional work always depend only on initial and final positions
Correct Answer: Electrostatic work is path independent; frictional work generally depends on path length
Explanation: Electrostatic force produced by fixed charges is conservative. Its work between two points is the same for all paths connecting those points. Friction is a common example of a non-conservative force because its work usually depends on the actual path length and surface contact. If an object travels a longer rough path, friction usually does more negative work. Electrostatic force is different because a round trip in an electrostatic field gives zero net work. Friction does not generally have this closed-path zero-work property. This distinction is central to why electrostatic potential energy can be defined, while ordinary frictional energy loss is not represented by a simple position-only potential energy.
10. A charge is moved in an electrostatic field from \(A\) to \(B\), then from \(B\) to \(C\), and finally from \(C\) to \(A\). If the electrostatic work values for the first two parts are \(3\,\text{J}\) and \(4\,\text{J}\), what is the work done by the electrostatic force from \(C\) to \(A\)?
ⓐ. \(+1\,\text{J}\)
ⓑ. \(-1\,\text{J}\)
ⓒ. \(+7\,\text{J}\)
ⓓ. \(-7\,\text{J}\)
Correct Answer: \(-7\,\text{J}\)
Explanation: \(\textbf{Known parts:}\) \(W_{A\to B}=3\,\text{J}\) and \(W_{B\to C}=4\,\text{J}\).
\(\textbf{Unknown part:}\) Let \(W_{C\to A}\) be the work for the last part.
\(\textbf{Closed path condition:}\) The path \(A\to B\to C\to A\) starts and ends at \(A\), so it is a closed path.
\(\textbf{Electrostatic rule:}\) Net work by electrostatic force over a closed path is \(0\).
\(\textbf{Equation:}\) \(3+4+W_{C\to A}=0\)
\(\textbf{Simplification:}\) \(7+W_{C\to A}=0\).
\(\textbf{Result:}\) \(W_{C\to A}=-7\,\text{J}\).
\(\textbf{Final result:}\) The work done from \(C\) to \(A\) is \(-7\,\text{J}\).
11. Assertion: Work done by electrostatic force in moving a charge between two fixed points is independent of the path followed.
Reason: Electrostatic force is a conservative force.
Which option is valid?
ⓐ. Assertion is true, but Reason is false
ⓑ. Assertion is false, but Reason is true
ⓒ. Both Assertion and Reason are true, and Reason explains Assertion
ⓓ. Both Assertion and Reason are true, but Reason does not explain Assertion
Correct Answer: Both Assertion and Reason are true, and Reason explains Assertion
Explanation: A conservative force is defined by the property that its work between two points does not depend on the path taken. Electrostatic force due to fixed charges has this conservative nature. Therefore, when a charge moves from one point to another in such a field, all possible paths give the same electrostatic work. This is not a separate accidental property; it follows directly from the conservative character of the force. The same idea also gives zero work over any closed path. Because the Reason states the principle responsible for the Assertion, it explains the Assertion properly.
12. A student claims that electrostatic work between two points must be larger for a longer path because more distance is covered. Which response best corrects the claim?
ⓐ. Distance alone decides electrostatic work, so the claim is valid
ⓑ. Only the endpoints decide electrostatic work, so the claim is not valid
ⓒ. The work is always positive for a longer path, so the claim is valid
ⓓ. The work is always zero for any open path, so the claim is not valid
Correct Answer: Only the endpoints decide electrostatic work, so the claim is not valid
Explanation: The claim confuses electrostatic force with forces whose work depends on the actual path. In an electrostatic field of fixed charges, work done by the electrostatic force is independent of path. A longer route and a shorter route between the same two points give the same work. The work is not automatically larger just because the distance travelled is larger. It is also not always zero for an open path; it can be positive, negative, or zero depending on the endpoints and charge. The special zero-work result applies to a complete closed path. Therefore, the correct correction is that only the endpoints decide the electrostatic work.
13. In an electrostatic field, a charge is taken from \(A\) to \(B\) along a path for which the electrostatic force does \(-6\,\text{J}\) of work. Along another path from \(A\) to \(B\), what is the change in potential energy of the charge-field system?
ⓐ. \(-6\,\text{J}\)
ⓑ. \(+6\,\text{J}\)
ⓒ. \(0\,\text{J}\)
ⓓ. \(+12\,\text{J}\)
Correct Answer: \(+6\,\text{J}\)
Explanation: \(\textbf{Given work by field:}\) \(W_{\text{field}}=-6\,\text{J}\) for motion from \(A\) to \(B\).
\(\textbf{Path idea:}\) Electrostatic work from \(A\) to \(B\) is the same along every path.
\(\textbf{Energy relation:}\) For electrostatic force, \(\Delta U=-W_{\text{field}}\).
\(\textbf{Substitution:}\) \(\Delta U=-(-6\,\text{J})\).
\(\textbf{Simplification:}\) \(\Delta U=+6\,\text{J}\).
\(\textbf{Meaning:}\) Negative work by the electrostatic force means the potential energy increases.
\(\textbf{Path check:}\) Choosing another path from \(A\) to \(B\) does not change the result because electrostatic force is conservative.
\(\textbf{Final result:}\) The change in potential energy is \(+6\,\text{J}\).
14. Which condition would fail if electrostatic force were not conservative?
ⓐ. Path-independent work between fixed points
ⓑ. Existence of force on a charge in an electric field
ⓒ. Action at a distance between electric charges
ⓓ. Measurement of work in \(\text{J}\)
Correct Answer: Path-independent work between fixed points
Explanation: The conservative nature of electrostatic force is what makes path-independent work possible. If electrostatic force were not conservative, the work between the same two points could depend on the route taken. Then a unique potential energy difference between those points could not be defined from position alone. A force might still act, and work could still be measured in \(\text{J}\), but the work would not have the special endpoint-only property. Action at a distance is also not the same idea as conservativeness. The essential feature that would fail is the unique path-independent value of work between fixed points.
15. A charge completes two different closed loops in the same electrostatic field. Loop \(1\) is small and loop \(2\) is large. Which comparison of electrostatic work is correct?
ⓐ. Work is greater for loop \(2\) because the loop is larger
ⓑ. Work is greater for loop \(1\) because the loop is smaller
ⓒ. Work is zero for loop \(1\) but non-zero for loop \(2\)
ⓓ. Work is zero for both loops
Correct Answer: Work is zero for both loops
Explanation: \(\textbf{Situation:}\) The charge completes a closed loop in each case.
\(\textbf{Important point:}\) A closed loop means the final position is the same as the initial position.
\(\textbf{Electrostatic principle:}\) Electrostatic force is conservative, so the total work done over any closed path is \(0\).
\(\textbf{Loop size check:}\) The result does not depend on whether the loop is small, large, circular, rectangular, or irregular.
\(\textbf{Mathematical form:}\) \(\oint \vec{F}_{\text{electrostatic}}\cdot d\vec{l}=0\)
\(\textbf{Conclusion:}\) Both loops have zero net work by the electrostatic force.
\(\textbf{Final result:}\) Work is zero for both loops.
16. A charge is moved slowly by an external agent from \(M\) to \(N\) in an electrostatic field. The external agent does \(+9\,\text{J}\) of work, and the kinetic energy of the charge does not change. What is the work done by the electrostatic force?
ⓐ. \(+9\,\text{J}\)
ⓑ. \(-9\,\text{J}\)
ⓒ. \(0\,\text{J}\)
ⓓ. \(+18\,\text{J}\)
Correct Answer: \(-9\,\text{J}\)
Explanation: \(\textbf{Given:}\) Work done by the external agent is \(W_{\text{ext}}=+9\,\text{J}\).
\(\textbf{Motion condition:}\) The charge is moved slowly, and its kinetic energy does not change.
\(\textbf{Work-energy idea:}\) If \(\Delta K=0\), then the net work on the charge is \(0\).
\(\textbf{Forces doing work:}\) The external agent and the electrostatic force are the relevant work contributions.
\(\textbf{Equation:}\) \(W_{\text{ext}}+W_{\text{field}}=0\)
\(\textbf{Substitution:}\) \(9\,\text{J}+W_{\text{field}}=0\).
\(\textbf{Solving:}\) \(W_{\text{field}}=-9\,\text{J}\).
\(\textbf{Energy meaning:}\) The external work is stored as increased electrostatic potential energy when kinetic energy remains unchanged.
\(\textbf{Final result:}\) The electrostatic force does \(-9\,\text{J}\) of work.
17. A positive charge is slowly moved by an external force from point \(A\) to point \(B\) in an electrostatic field. Its kinetic energy remains unchanged. If the electrostatic force does \(-12\,\text{J}\) of work, what is the work done by the external force?
ⓐ. \(-12\,\text{J}\)
ⓑ. \(+12\,\text{J}\)
ⓒ. \(0\,\text{J}\)
ⓓ. \(+24\,\text{J}\)
Correct Answer: \(+12\,\text{J}\)
Explanation: \(\textbf{Given:}\) Work done by the electrostatic force is \(W_{\text{field}}=-12\,\text{J}\).
\(\textbf{Motion condition:}\) The charge is moved slowly, so there is no change in kinetic energy.
\(\textbf{Work-energy principle:}\) If \(\Delta K=0\), the total work done on the charge must be \(0\).
\(\textbf{Work balance:}\) The external force and electrostatic force together do zero net work.
\(\textbf{Equation:}\) \(W_{\text{ext}}+W_{\text{field}}=0\)
\(\textbf{Substitution:}\) \(W_{\text{ext}}+(-12\,\text{J})=0\).
\(\textbf{Solving:}\) \(W_{\text{ext}}=+12\,\text{J}\).
\(\textbf{Energy meaning:}\) The external work is stored as increased electrostatic potential energy.
\(\textbf{Final result:}\) The work done by the external force is \(+12\,\text{J}\).
18. Which statement correctly connects electrostatic work and electrostatic potential energy?
ⓐ. If electrostatic force does positive work, potential energy increases
ⓑ. If electrostatic force does negative work, potential energy decreases
ⓒ. Potential energy change equals the negative of electrostatic work
ⓓ. Potential energy change is always zero for an open displacement
Correct Answer: Potential energy change equals the negative of electrostatic work
Explanation: Electrostatic force is conservative, so its work can be connected with potential energy. The relation is \(\Delta U=-W_{\text{field}}\), where \(W_{\text{field}}\) is the work done by the electrostatic force. If the electrostatic force does positive work, the potential energy decreases. If the electrostatic force does negative work, the potential energy increases. The energy change is not always zero for an open displacement because the initial and final points may have different potential energies. The negative sign shows that work done by the field comes from stored electrostatic potential energy. This relation is one of the main reasons electrostatic interactions can be treated using energy methods.
19. A charge is moved from point \(P\) to point \(Q\). During this motion, the electrostatic potential energy changes by \(+15\,\text{J}\). What work is done by the electrostatic force?
ⓐ. \(+15\,\text{J}\)
ⓑ. \(0\,\text{J}\)
ⓒ. \(-15\,\text{J}\)
ⓓ. \(+30\,\text{J}\)
Correct Answer: \(-15\,\text{J}\)
Explanation: \(\textbf{Given:}\) Change in electrostatic potential energy is \(\Delta U=+15\,\text{J}\).
\(\textbf{Required:}\) Work done by the electrostatic force, \(W_{\text{field}}\).
\(\textbf{Energy relation:}\) For electrostatic force, \(\Delta U=-W_{\text{field}}\).
\(\textbf{Rearranging:}\) \(W_{\text{field}}=-\Delta U\).
\(\textbf{Substitution:}\) \(W_{\text{field}}=-(+15\,\text{J})\).
\(\textbf{Simplification:}\) \(W_{\text{field}}=-15\,\text{J}\).
\(\textbf{Sign meaning:}\) The field does negative work because potential energy is being increased.
\(\textbf{Final result:}\) The electrostatic force does \(-15\,\text{J}\) of work.
20. A charge is slowly displaced by an external agent in an electrostatic field. Which condition tells that the external work is stored completely as electrostatic potential energy?
ⓐ. The charge moves along a longer curved path
ⓑ. The charge has no change in kinetic energy
ⓒ. The electrostatic force remains zero everywhere
ⓓ. The external force acts perpendicular to motion
Correct Answer: The charge has no change in kinetic energy
Explanation: When a charge is moved slowly, its kinetic energy does not change appreciably. In that case, the work done by the external agent is not converted into kinetic energy. Instead, it changes the electrostatic potential energy of the charge-field system. The external force balances the electrostatic force at each stage for slow movement. The path may be straight or curved, but path length is not the key condition for storing energy. If the kinetic energy changed, part of the external work would appear as kinetic energy. Therefore, unchanged kinetic energy allows external work to be identified with the change in electrostatic potential energy.
