The Electronvolt

• Work-Energy Theorem: The work done $W$ on a charge $q$ by an electric field is given by the product of the charge and the potential difference $\Delta V$ it moves through, expressed as $W = q\Delta V$.

• Derivation: By substituting the charge of an electron ($e = 1.602 \times 10^{-19}$ C) and a potential difference of $1$ V into the formula, we find that $1 \text{ eV} = (1.602 \times 10^{-19} \text{ C}) \times (1 \text{ V}) = 1.602 \times 10^{-19} \text{ J}$.

• Kinetic Energy Conversion: When a particle is accelerated from rest, the work done by the field is entirely converted into kinetic energy. Therefore, a particle with charge $e$ accelerated through $V$ volts will have a kinetic energy of $E_k = eV$ in electronvolts.

• From eV to Joules: To convert an energy value from electronvolts to Joules, multiply the value by the elementary charge constant $1.602 \times 10^{-19}$. This is necessary whenever using standard SI formulas like $E = hf$ or $E = \frac{1}{2}mv^2$ where other units are in SI.

• From Joules to eV: To convert from Joules to electronvolts, divide the energy value by $1.602 \times 10^{-19}$. This is typically done at the end of a calculation to express a result in a more readable format for atomic scales.

• Decision Criteria: Always perform intermediate calculations in Joules to ensure consistency with other SI units (like kilograms for mass or meters per second for velocity). Only convert to eV for the final presentation or when comparing to atomic energy levels.

Comparison of Energy Units and Potential

• Energy vs. Potential: A common mistake is treating the electronvolt as a unit of voltage. While they are numerically related in specific scenarios, the eV measures the 'capacity to do work' while the Volt measures 'potential energy per unit charge'.

• Mass-Energy Equivalence: In high-energy physics, the eV is often used to measure mass via Einstein's equation $E=mc^2$. Mass is frequently expressed in units of $\text{eV}/c^2$, which simplifies calculations in particle physics.

• Check the Magnitude: If your energy value in Joules is a large number (e.g., $> 1$), it is almost certainly not an atomic-scale energy. Conversely, energy in eV should usually be a 'human-readable' number like $2.5$ eV or $13.6$ eV for atomic transitions.

• Prefix Awareness: Be prepared to handle prefixes such as keV ($10^3$ eV), MeV ($10^6$ eV), and GeV ($10^9$ eV). These are standard in nuclear physics and particle acceleration problems.

• Formula Consistency: When using the photoelectric equation $hf = \Phi + E_{k,max}$, ensure all three terms are in the same unit. If the work function $\Phi$ is given in eV, you must convert it to Joules before adding it to a kinetic energy calculated in Joules.

• The 'Charge' Trap: Students often forget that the eV unit already 'contains' the charge of the electron. If a question asks for the energy of an alpha particle (charge $+2e$) accelerated through $100$ V, the energy is $200$ eV, not $100$ eV.

• Inverse Conversion: A frequent error is multiplying by $1.6 \times 10^{-19}$ when one should divide. Remember that since the Joule is a much larger unit, the numerical value in eV must be much larger than the numerical value in Joules.

• Unit Confusion: Do not confuse the symbol 'eV' with the variables $e$ (charge) and $V$ (voltage) in the formula $W=qV$. The unit 'eV' is a single entity representing a specific quantity of energy.