(e) energy transfer eV = 1/2 mv² for electrons and other charged particles.

The idea

When a charged particle is accelerated through a potential difference V, the electrical work done equals the kinetic energy gained. For a particle of charge q, the energy transferred is qV. In many problems the charge is the elementary charge e, so the energy transferred is eV. If the particle starts from rest, this energy becomes kinetic energy, so eV = ½mv², where m is the mass and v is the final speed. This equation links speed and accelerating voltage. For example, an electron accelerated through 100 V gains 100 eV, which is 1.60 × 10⁻¹⁷ J; using ½mv² gives a speed of about 5.9 × 10⁶ m s⁻¹. Remember that eV is an energy unit: 1 eV = 1.60 × 10⁻¹⁹ J.

What earns the marks

• Electrical energy transferred to a charged particle accelerated through potential difference V is qV, or eV when the charge magnitude is the elementary charge e.
• For a particle starting from rest, the energy transferred equals the kinetic energy gained, so eV = ½mv².
• The electronvolt is an energy unit: 1 eV = 1.60 × 10⁻¹⁹ J.
• Use the correct mass and charge for the particle: for an electron, m = 9.11 × 10⁻³¹ kg and e = 1.60 × 10⁻¹⁹ C.
• When using eV = ½mv² with mass in kg and speed in m s⁻¹, you must convert the energy in eV to joules before calculating.

Common mistakes and how to put them right

• Using the electron mass for a proton or other charged particle.

  Put it right: Use the mass of the specific particle named in the question.

• Forgetting to convert electronvolts to joules before substituting into ½mv².

  Put it right: Multiply eV by 1.60 × 10⁻¹⁹ J eV⁻¹ to obtain energy in joules.

• Assuming the particle starts from rest when it may have an initial speed.

  Put it right: Check the wording; if it does not start from rest, include the initial kinetic energy.

• Confusing potential difference V with speed v.

  Put it right: Keep the symbols distinct and check which quantity the question gives.

Exam tips

• Write the energy equation first, then substitute values with units; this makes the conversion from eV to J clear.
• Check whether the question asks for speed, kinetic energy or accelerating voltage, and rearrange before calculating.
• Keep at least three significant figures in intermediate values and round only the final answer.
• If the particle is not an electron, look up or use the given charge and mass rather than assuming electron values.