Conservation of Electric Energy

When a charged particle moves through an electric field, work is done either on the particle or by the particle. This exchange of energy obeys the same fundamental principle that governs all of physics: energy is never created or destroyed, only transformed. In electrostatic systems, this means that the total mechanical energy — kinetic energy plus electric potential energy — remains constant, provided no non-conservative forces (like friction) are present. This principle is called conservation of electric energy. Electric potential energy is stored in the configuration of charges, much like gravitational potential energy is stored when you lift a ball above the ground. When a positive charge moves from a region of high electric potential to low electric potential, it loses potential energy and gains kinetic energy — exactly analogous to a ball rolling downhill. Conversely, pushing a positive charge toward another positive charge requires doing work against the repulsive force, storing energy in the system. Understanding energy conservation in electric fields is essential for analysing circuits, particle accelerators, and biological ion transport. It allows physicists and engineers to predict the speed of charged particles, determine the voltage needed to accelerate an electron to a given energy, and explain why lightning bolts carry so much destructive power. The mathematics is elegant: set the total energy at one point equal to the total energy at another, and solve.

Key takeaways
  • Electric Potential Energy (U)
  • The Work-Energy Theorem in Electric Fields
  • Conservation of Total Mechanical Energy
  • The Electron-Volt: A Practical Energy Unit
  • Energy Stored in a Capacitor

Electric Potential Energy (U)

The electric potential energy of a charge q at a point in space depends on the electric potential V at that point: U = qV. For two point charges q₁ and q₂ separated by distance r, the potential energy of the pair is U = kq₁q₂/r, where k = 8.99 × 10⁹ N·m²/C² is Coulomb's constant. This energy is positive when charges repel (same sign) and negative when they attract (opposite signs).

The Work-Energy Theorem in Electric Fields

The work done by the electric force on a charge moving from point A to point B equals the decrease in electric potential energy: W = U_A − U_B = q(V_A − V_B). This work converts potential energy into kinetic energy. If the charge speeds up, potential energy decreased; if it slows down, potential energy increased. The quantity (V_A − V_B) is the voltage difference, measured in volts (V = J/C).

Conservation of Total Mechanical Energy

In a system with no non-conservative forces, the total energy E_total = KE + U_electric is constant. Writing this out: ½mv_A² + qV_A = ½mv_B² + qV_B. This single equation lets you find the speed of a particle at any point if you know the potential there. This is the electric analogue of the roller-coaster energy equation from mechanics — potential hills and valleys correspond to regions of high and low electric potential.

The Electron-Volt: A Practical Energy Unit

Because Joules are inconveniently large for subatomic particles, physicists use the electron-volt (eV). One eV is the kinetic energy gained by an electron (charge e = 1.602 × 10⁻¹⁹ C) accelerated through a potential difference of exactly 1 V: 1 eV = 1.602 × 10⁻¹⁹ J. A particle gun accelerating electrons through 10,000 V gives each electron 10,000 eV = 10 keV of kinetic energy. Particle accelerators at CERN push protons to energies of trillions of eV (TeV).

Energy Stored in a Capacitor

A capacitor with capacitance C charged to voltage V stores electric potential energy U = ½CV². This energy resides in the electric field between the plates. When the capacitor discharges through a circuit, this stored energy is released — driving current, producing light in a flash bulb, or powering a defibrillator pulse. The factor of ½ arises because charging requires doing work against the growing field: as charge accumulates, the voltage rises, making each additional increment of charge harder to add.

The Key Equation

12mvA2+qVA=12mvB2+qVB\frac{1}{2}mv_A^2 + qV_A = \frac{1}{2}mv_B^2 + qV_B
  • m — Mass of charged particle (kg): The inertial mass of the particle moving through the field.
  • v_A,\; v_B — Speed at points A and B (m/s): The particle's speed at the initial point A and final point B.
  • q — Electric charge (C (coulombs)): The charge of the particle; positive for protons, negative for electrons.
  • V_A,\; V_B — Electric potential at A and B (V (volts)): The electric potential at each location, determined by the source charges creating the field.

Worked Example

Problem. An electron (mass 9.11 × 10⁻³¹ kg, charge −1.602 × 10⁻¹⁹ C) starts from rest at a negative plate and is accelerated through a potential difference of ΔV = 500 V toward a positive plate. What is the electron's speed when it reaches the positive plate?

vB=2∣q∣ΔVm≈1.33×107 m/sv_B = \sqrt{\frac{2|q|\Delta V}{m}} \approx 1.33 \times 10^7 \text{ m/s}

Answer. The electron reaches a speed of approximately 1.33 × 10⁷ m/s — about 4.4% the speed of light.

Figure
Diagram
Energy Conservation: Electron Accelerated Between Plates− plate+ plateV=0 VV=500 VUniform electric field →e⁻v₀=0e⁻v=1.33×10⁷m/sPoint A (left)KE = 0U = qV = highPoint B (right)KE = 8.01×10⁻¹⁷ JU = qV = lowKE_A + U_A = KE_B + U_B (Total energy conserved)
Origins · 1 minAlessandro Volta and how this was discovered
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Alessandro Volta
1745–1827 · Italian

Volta invented the electric battery (voltaic pile) in 1800, providing for the first time a steady source of electric potential difference. This allowed sustained study of how charges move through potential differences and how electrical energy is stored and transferred. The unit of electric potential, the volt (V), is named in his honour. Volta was so famous in his lifetime that Napoleon Bonaparte personally attended his lectures at the Institut de France and awarded him a gold medal. Napoleon was so impressed he made Volta a Count and senator of the Kingdom of Italy.

  • Electric Potential and Voltage — Electric potential V is the quantity that determines electric potential energy (U = qV), so understanding potential is prerequisite to applying energy conservation.
  • Coulomb's Law — The electric potential energy between two point charges (U = kq₁q₂/r) is derived directly from Coulomb's law, linking force and energy in electrostatics.
  • Conservation of Mechanical Energy — The same KE + PE = constant principle from mechanics is extended here to include electric potential energy, making this a direct application of a broader conservation law.
  • Capacitors and Energy Storage — Capacitors store electric energy (U = ½CV²) and release it, making them a direct practical application of energy conservation in electrostatic systems.