BPSC 70th CCE Re-exam Prelims General Studies · Question 90 of 142

Correct — D, eV. Potential difference is defined as work per unit charge, V = W/q, so moving a charge q through a potential difference V does work W = qV on it. Here the charge is e coulomb and the potential difference is V volts, so the work done is eV joules. Because the electron starts from rest it has no initial kinetic energy, and by the work-energy theorem the whole of that work appears as kinetic energy: final energy = eV. That is the entire derivation, and the mass never enters it. Mass enters only if you go one step further and ask for the speed: setting the kinetic energy ½Mv² equal to eV and solving gives v = √(2eV/M). The setter has put that half-remembered formula in the option list on purpose. Dimensional analysis alone kills the other three without any physics. Coulomb × volt = joule, so only (d) has the dimensions of energy at all. A coulomb per volt is a farad, the unit of capacitance, so (a) e/V is a capacitance. (c) eV/M is joules per kilogram, which is energy per unit mass — the same dimensions as velocity squared, and in fact eV/M equals ½v², the kinetic energy carried by each kilogram rather than the energy the question asks for. (b) MeV multiplies energy by a mass and gives joule-kilograms. Note the typography: in the booklet (A) and (C) are printed as stacked fractions, e over V and eV over M, which our transcription flattens to e/V and eV/M — (C) eV/M and (D) eV are genuinely different options and not two renderings of the same thing. Because (d) is exactly right, that printing quirk does not damage the question.

• (a)e/V — Printed in the booklet as e over V. Dividing charge by potential difference gives coulombs per volt, which is the farad — the unit of capacitance, not of energy. The option exists for the candidate who remembers that e and V belong together in the formula but not which way round; the definition V = W/q rearranges to W = qV, a product, never a quotient.
• (b)MeV — A double trap. Taken as algebra it is M × e × V, mass times charge times potential, whose units are kilogram-joules — not an energy. Taken as a symbol, 'MeV' is the standard particle-physics abbreviation for megaelectronvolt, which genuinely is an energy, so a candidate who reads the three characters instead of the algebra can convince himself this is the right dimensional shape.
• (c)eV/M — Printed as eV over M, and the strongest wrong answer. It is the trap for anyone who reaches for the speed formula v = √(2eV/M) and stops halfway: eV/M is ½v², the energy per unit mass in joules per kilogram, a specific energy. The stem asks for the final energy in joules, not the energy per kilogram, and dividing by the mass is exactly the step the work-energy theorem does not require.

Electric potential is defined as potential energy per unit charge, and the volt is defined so that one joule of work is done in moving one coulomb through one volt. Everything on this card follows from that single definition. A charge q released from rest and allowed to fall through a potential difference V has work qV done on it by the electric field, and since the field is conservative that work is exactly the loss of electrostatic potential energy, which reappears as kinetic energy. It is the electrical version of a stone falling through a height h and arriving with kinetic energy mgh — the field does the work, the body picks up the speed, and the identity of the body only matters if you want the speed rather than the energy. This is also the origin of the electronvolt as a unit: one electronvolt is defined as the energy an electron gains falling through one volt. Since the 2019 redefinition of the SI fixed the elementary charge at exactly 1.602176634 × 10⁻¹⁹ coulomb, one electronvolt is exactly 1.602176634 × 10⁻¹⁹ joule. Physicists quote particle energies in eV, keV, MeV, GeV and TeV precisely because those numbers are the accelerating voltages, which is why an 'X-ray tube run at 100 kV' and 'X-rays of up to 100 keV' say the same thing twice.

Two habits solve every question of this shape. The first is to check units before doing any physics: write out coulomb × volt = joule and the answer is already isolated, since none of the other three combinations is an energy. Candidates who cannot recall whether the mass belongs in the formula can still reach (d) this way in about ten seconds, which is what makes dimensional analysis worth practising as a technique rather than as a topic. The second is to notice precisely what the question asks for — energy or speed. Energy is mass-independent: a proton and an electron accelerated through the same 1,000 volts both acquire 1,000 eV. Speed is not: v = √(2eV/M), so the far lighter electron ends up about 43 times faster than the proton. Numerically, an electron falling through 1 volt reaches about 5.9 × 10⁵ metres per second, roughly 593 km/s, and through 10,000 volts about 5.9 × 10⁷ m/s, which is nearly a fifth of the speed of light — at which point the Newtonian formula starts to under-report the speed and a relativistic treatment is needed. The energy expression eV, however, remains exact at every voltage, because it comes from the definition of the volt and not from any model of how mass and speed are related.

• Potential difference is work per unit charge, V = W/q, so work done on a charge q crossing V is W = qV; for the electron here, W = eV joules.
• The work-energy theorem gives final kinetic energy minus initial kinetic energy = work done; the electron starts from rest, so its final energy is the whole of eV.
• One electronvolt is defined as the energy gained by an electron accelerated through one volt, and equals exactly 1.602176634 × 10⁻¹⁹ joule since the SI redefinition of 20 May 2019 fixed the elementary charge.
• Mass enters only through the speed: v = √(2eV/M). An electron of mass about 9.11 × 10⁻³¹ kg accelerated through 1 volt reaches roughly 5.9 × 10⁵ m/s.
• Dimensions of the four options: coulomb × volt = joule (energy); coulomb ÷ volt = farad (capacitance); joule ÷ kilogram = specific energy, equal to velocity squared; joule × kilogram is not a physical quantity in ordinary use.

• Importing the mass because the stem mentions it. The stem gives M so that the wrong options can use it; the energy expression does not contain it and never did.
• Confusing 'MeV' the algebraic product with MeV the megaelectronvolt. The exam is testing the algebra, not the symbol.
• Answering with eV/M when the question asks for energy. That is the specific energy in joules per kilogram — the same quantity as ½v², one step short of the speed formula.

BPSC's science is one formula, one substitution, four expressions — often with the letters supplied in the stem so no numbers are needed at all, and on this re-examination of January 2025 the answer is simply the product of the two symbols given. UPSC almost never sets a bare formula; it asks where the physics shows up in the world, as with the 2009 question on accelerating sub-atomic particles to near light speed, so the same knowledge has to be carried in an applied form.

• practice — not a real PYQ

  One electronvolt is equal to approximately

  • (a)1.6 × 10⁻¹⁹ joule
  • (b)1.6 × 10⁻¹⁹ watt
  • (c)9.1 × 10⁻³¹ joule
  • (d)6.6 × 10⁻³⁴ joule

  Answer(a) 1.6 × 10⁻¹⁹ joule — exactly 1.602176634 × 10⁻¹⁹ J since the elementary charge was fixed by definition in 2019. The value 9.1 × 10⁻³¹ is the electron's mass in kilograms and 6.6 × 10⁻³⁴ is Planck's constant in joule-seconds.

• practice — not a real PYQ

  A proton and an electron are each accelerated from rest through the same potential difference. Which of the following is true?

  • (a)Both gain the same kinetic energy and the same speed
  • (b)Both gain the same kinetic energy, but the electron ends up faster
  • (c)Both reach the same speed, but the proton gains more kinetic energy
  • (d)The proton gains more kinetic energy and also moves faster

  Answer(b) Both gain the same kinetic energy, but the electron ends up faster — the energy qV depends only on the charge, which is equal in magnitude for both, while the speed √(2qV/m) depends on the mass, and the proton is about 1,836 times heavier.