For an electron moving in a direction opposite to the electric field

Question

BlackTom AI · Accepted Answer

An electron has negative charge, so the force it experiences is opposite the electric field: F = qE = −eE. 1) Relevant concepts - Electric field and potential: E = −dV/dx (in one dimension) and more generally E = −∇V. Moving opposite the field corresponds to displacement dℓ opposite to E. - Electric potential change: ΔV = −∫ E · dℓ. - Electric potential energy: U = qV, so ΔU = q ΔV. - Work by the field: W_field = −ΔU (positive work by the field decreases potential energy). 2) Step-by-step reasoning - Choose a uniform field for clarity: let E point to the +x direction. "Opposite to the electric field" means the electron moves toward −x (dℓ is opposite to E). - Compute ΔV: ΔV = −∫ E · dℓ = −E(Δx). If Δx is negative (moving left), then ΔV = −E(negative) = +E|Δx|, so ΔV > 0. Thus the electric potential increases along this displacement. - Compute ΔU: ΔU = q ΔV. For an electron q = −e, so ΔU = −e (ΔV). Since ΔV > 0, ΔU < 0: the electron’s electric potential energy decreases. - Work/kinetic energy check: The force on the electron is F = −eE which points opposite E and therefore along the electron’s displacement. The field does positive work on the electron, increasing its kinetic energy while decreasing its potential energy, consistent with ΔU < 0. 3) Intermediate expressions shown - ΔV = −E Δx (with Δx < 0) ⇒ ΔV > 0. - ΔU = q ΔV = (−e)(ΔV) ⇒ ΔU < 0. 4) Why the correct answer is correct Because moving opposite the field raises the scalar potential V (ΔV > 0), and since the electron has negative charge, U = qV decreases (ΔU < 0). 5) Why other choices are incorrect - "U decreases and V decreases": contradicts ΔV > 0 derived from ΔV = −∫E·dℓ. - "U increases and V increases": wrong sign for ΔU (q is negative). - "U increases and V decreases": both signs reversed. - "Both remain constant": there is nonzero work by/on the field so quantities change. 6) Conclusion Therefore the electron’s potential energy decreases while its electric potential increases, matching the provided correct answer.

For an electron moving in a direction opposite to the electric fieldSingle choice

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