Capacitance

Capacitance is a fundamental concept that quantifies the ability of a system of conductors to store electric charge for a given potential difference. The concept appears in every branch of electrical engineering, from filter design to energy storage systems. In electromagnetics, capacitance is analyzed rigorously for different geometries including parallel plates, coaxial cylinders, and concentric spheres, giving formulas that are directly testable in GATE.

Core Concept: Definition and Physical Meaning

Capacitance C is defined as C = Q / V, where Q is the total charge stored on one conductor and V is the potential difference between the two conductors. The unit of capacitance is the Farad (F), and practical values range from picofarads in integrated circuits to millifarads in power capacitors. The definition implies that a higher capacitance system can hold more charge at the same voltage.

The capacitance of any geometry is determined entirely by the physical dimensions and the permittivity of the medium between the conductors. It does not depend on Q or V individually, only on their ratio. This is because the electric field inside any capacitor scales linearly with Q, and the potential difference scales linearly with E, so V also scales linearly with Q, making C = Q/V a constant that depends only on geometry and material.

Mathematical Expression

The parallel plate capacitor formula is C = epsilon * A / d. Here A is the plate area, d is the separation, and epsilon = epsilon_r * epsilon_0 is the permittivity of the medium. The field between the plates is uniform and equals E = sigma / epsilon, where sigma = Q/A is the surface charge density. The potential difference is V = E * d, leading directly to C = Q/V = epsilon*A/d.

For the coaxial cylindrical capacitor with inner radius a, outer radius b, and length L, the field between the cylinders is radial and given by E = rho_L / (2*pi*epsilon*r), where rho_L is the charge per unit length. Integrating E from a to b gives V = (rho_L / 2*pi*epsilon) * ln(b/a). Since Q = rho_L * L, capacitance C = 2*pi*epsilon*L / ln(b/a).

For the spherical capacitor with inner radius a and outer radius b, the radial field is E = Q / (4*pi*epsilon*r^2). Integrating from a to b gives V = Q*(b-a) / (4*pi*epsilon*a*b). So C = 4*pi*epsilon*a*b / (b-a). If the outer sphere is removed to infinity (b approaching infinity), C reduces to 4*pi*epsilon*a, which is the capacitance of an isolated sphere.

Practical Understanding

Capacitance analysis is used to design coaxial cables where the ratio ln(b/a) determines the capacitance per unit length and affects signal propagation. In integrated circuit design, the parallel plate formula governs MOS capacitor behavior. Spherical capacitance is relevant in electrostatic shielding problems. For multi-dielectric capacitors, layers in series (normal to field) add as 1/C_total = sum of 1/C_i, while parallel layers add directly: C_total = sum of C_i.

Given:
Coaxial capacitor: a = 2 mm = 0.002 m, b = 6 mm = 0.006 m, L = 0.5 m
Medium: epsilon_r = 4, epsilon_0 = 8.85e-12 F/m

Why this formula applies:
Coaxial geometry with radial field: C = 2*pi*epsilon*L / ln(b/a)

Formula:
C = 2 * pi * epsilon_r * epsilon_0 * L / ln(b/a)

Substitution:
epsilon = 4 * 8.85e-12 = 3.54e-11 F/m
ln(b/a) = ln(0.006/0.002) = ln(3) = 1.0986
C = 2 * 3.14159 * 3.54e-11 * 0.5 / 1.0986

Calculation:
Numerator = 2 * 3.14159 * 3.54e-11 * 0.5 = 1.112e-10
C = 1.112e-10 / 1.0986 = 1.012e-10 F

Final Answer:
C = 101.2 pF approximately

Exam Tip: For GATE, always note that capacitance depends only on geometry and medium, not on the charge or voltage. For coaxial capacitor, ln(b/a) is the key term. If b/a = e (Euler's number), then ln(b/a) = 1, simplifying the formula directly to C = 2*pi*epsilon*L.

• C = Q/V, determined entirely by geometry and medium permittivity.
• Parallel plate: C = epsilon*A/d. Larger area or smaller separation increases C.
• Coaxial: C = 2*pi*epsilon*L / ln(b/a). Key term is the logarithm of radius ratio.
• Spherical: C = 4*pi*epsilon*ab/(b-a). Isolated sphere: C = 4*pi*epsilon*a.
• Series combination: 1/C_total = 1/C1 + 1/C2 (same Q on each).
• Parallel combination: C_total = C1 + C2 (same V across each).

Quick Revision

• C = Q/V (Farads). Depends only on geometry and permittivity, not on Q or V alone.
• Parallel plate: C = epsilon*A/d.
• Coaxial: C = 2*pi*epsilon*L/ln(b/a).
• Spherical: C = 4*pi*epsilon*ab/(b-a).
• Inserting dielectric (epsilon_r > 1) increases C by factor epsilon_r.
• Series: reciprocals add. Parallel: capacitances add directly.
• GATE trap: Do not confuse ln(b/a) with (b-a)/a in coaxial formula.