Which of the following statements is false?

Question

BlackTom AI · Accepted Answer

Electric field E equals −∇V, so the field points from regions of higher electric potential toward regions of lower potential. For a positive charge q the force is F = qE = −q∇V, so the force (and motion under that force alone) is toward decreasing potential, not toward increasing potential. Therefore the statement "Positive charges move from low to high potential if acted on only by the electric field" is false.

Relevant concepts and definitions:
- Electric potential V and electric field E are related by E = −∇V (in one dimension E = −dV/dx). The negative sign means E points from higher V to lower V.
- Force on a point charge q is F = qE. For q > 0 the force direction is the same as E.
- Potential energy U of a charge is U = qV, so ΔU = qΔV. Work done by the electric field on the charge is W_field = −ΔU = −qΔV.

Step-by-step reasoning and intermediate steps:
1. From E = −∇V, if V decreases in a given direction (∇V negative), E points that way.
2. For q > 0, F = qE = −q∇V. Thus the force is toward regions of lower potential (decreasing V).
3. If a charge moves on an equipotential surface, ΔV = 0 so ΔU = qΔV = 0 and W_field = −qΔV = 0. Hence no work is done by the field along an equipotential and the potential energy does not change.

Why the other options are not the false statement:
- "The electric field is always perpendicular to equipotential surfaces" is true because any component of E along the surface would produce a potential change.
- "The work done by the field on a charge that moves on an equipotential surface is zero" follows from ΔV = 0 ⇒ W_field = −qΔV = 0.
- "The change in potential between two points on an equipotential surface is zero" is the definition of an equipotential.
- "The potential energy of a charge does not change if it moves on an equipotential surface" follows from ΔU = qΔV = 0.

Conclusion: The false statement is "Positive charges move from low to high potential if acted on only by the electric field," matching the provided answer.

Which of the following statements is false?单项选择题

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