Energy Stored in Electric Field

When charges are separated and placed on a capacitor or distributed in space, work is done against the electric forces to bring them to their positions. This work is stored as energy in the electric field. Unlike energy stored in mechanical springs or chemical bonds, electrostatic energy is stored in the field itself, distributed throughout the volume of space where the field exists. This viewpoint is fundamental to electromagnetic field theory and is a recurring GATE topic.

Core Concept: Where Does the Energy Come From

To charge a capacitor from zero to final charge Q, successive small increments of charge dq must be brought from the negative plate to the positive plate against an increasing potential difference V = q/C. The work done to bring charge dq is dW = V dq = (q/C) dq. Integrating from 0 to Q gives the total stored energy W = Q^2 / (2C). Using C = Q/V, this can also be written as W = (1/2) C V^2 = (1/2) Q V.

This energy does not reside on the plates themselves but in the electric field between the plates. For a parallel plate capacitor, the field is uniform and occupies the volume A*d between the plates. Substituting C = epsilon*A/d and V = E*d into W = (1/2)*C*V^2 and dividing by volume A*d gives the energy per unit volume as w_e = (1/2)*epsilon*E^2. This is called the electric energy density, and it is a local quantity that applies at every point in space where a field exists.

Mathematical Expression

The general expression for electric energy density at any point in space is w_e = (1/2) * D dot E = (1/2) * epsilon * E^2, where E is the magnitude of the electric field at that point and epsilon is the local permittivity. The total electrostatic energy stored in a volume V is found by integrating the energy density over the entire volume: W = integral over all space of (1/2) * epsilon * E^2 dv.

For a capacitor, the three equivalent forms of stored energy are W = (1/2) * C * V^2 = Q^2 / (2C) = (1/2) * Q * V. All three forms give the same result and are related through C = Q/V. The choice of which form to use depends on which quantities are known: use (1/2)CV^2 when voltage is given, Q^2/(2C) when charge is given, and the field integral form when the field distribution is known.

Practical Understanding

Energy stored in electric fields is important in practical applications including defibrillators (discharge of high-voltage capacitors), camera flash circuits, pulse power systems, and filter capacitors in power supplies. In all these cases, the energy W = (1/2)CV^2 is stored slowly during charging and released rapidly during discharge. The energy density formula also shows that a stronger dielectric (higher epsilon_r) stores more energy per unit volume for the same applied field, which is why high-permittivity ceramics are used in compact high-energy capacitors.

Given:
Parallel plate capacitor: A = 0.01 m2, d = 2 mm = 0.002 m
epsilon_r = 5, epsilon_0 = 8.85e-12 F/m, V = 1000 V

Why this formula applies:
Parallel plate: C = epsilon*A/d, then W = (1/2)*C*V^2
Also verify via energy density: E = V/d, w_e = (1/2)*epsilon*E^2

Formula:
C = epsilon_r * epsilon_0 * A / d
W = (1/2) * C * V^2

Substitution:
C = 5 * 8.85e-12 * 0.01 / 0.002 = 5 * 8.85e-12 * 5 = 2.2125e-10 F
W = 0.5 * 2.2125e-10 * (1000)^2

Calculation:
W = 0.5 * 2.2125e-10 * 1e6 = 1.106e-4 J

Verification via energy density:
E = 1000/0.002 = 5e5 V/m
w_e = 0.5 * 5 * 8.85e-12 * (5e5)^2 = 0.5 * 4.425e-11 * 2.5e11 = 5.53 J/m3
Volume = 0.01 * 0.002 = 2e-5 m3
W = 5.53 * 2e-5 = 1.106e-4 J (matches)

Final Answer:
W = 110.6 uJ, Energy density = 5.53 J/m3

Exam Tip: GATE often asks which expression to use for W. Remember all three are equivalent: W = (1/2)CV^2 = Q^2/2C = (1/2)QV. For distributed fields, use W = (1/2)*integral(epsilon*E^2 dv). A common trap is forgetting the factor of 1/2, which comes from the charging process, not just Q times V.

• Energy stored in capacitor: W = (1/2)CV^2 = Q^2/(2C) = (1/2)QV.
• Energy density: w_e = (1/2)*epsilon*E^2 = (1/2)*D*E (units: J/m3).
• Total energy: W = integral of (1/2)*epsilon*E^2 over all volume.
• Energy resides in the field, not on the conductors.
• Doubling E increases energy density by a factor of 4 (E squared relationship).

Quick Revision

• W = (1/2)CV^2 = Q^2/(2C) = (1/2)QV: all three are equivalent.
• Energy density: w_e = (1/2)*epsilon*E^2 = (1/2)*D.E, unit J/m3.
• Total W = volume integral of w_e over entire field region.
• Factor of 1/2 arises because V increases during charging (not constant).
• Higher epsilon_r stores more energy for same E field.
• GATE trap: W = QV (without 1/2) is wrong. Always W = (1/2)QV.
• Energy density scales as E squared: important for field concentration problems.