# Lecture 5: Capacitance and Dielectrics ## Physics II — Electromagnetism, Optics & Modern Physics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. Define capacitance and calculate it for parallel plate, cylindrical, and spherical capacitors 2. Determine the equivalent capacitance of capacitors in series and parallel combinations 3. Calculate the energy stored in a charged capacitor 4. Explain the effect of dielectric materials on capacitance and describe the physical mechanism 5. Solve circuit problems involving combinations of capacitors with and without dielectrics --- ## Lecture Content ### I. Capacitance A **capacitor** is a device that stores electric charge and energy. It consists of two conductors (plates) separated by an insulator, with one plate carrying charge +Q and the other carrying charge -Q. The **capacitance** is defined as the ratio of the stored charge to the potential difference across the plates: C = Q / Delta V. The SI unit of capacitance is the **farad** (F) = 1 C/V, though typical capacitors have capacitances measured in microfarads (uF), nanofarads (nF), or picofarads (pF). Capacitance depends only on the geometry of the device and the material between the plates, not on Q or V. For a **parallel plate capacitor** with plate area A and separation d, the electric field between the plates is E = sigma / epsilon_0 = Q / (epsilon_0 A), the potential difference is Delta V = Ed = Qd / (epsilon_0 A), and therefore the capacitance is C = epsilon_0 A / d. Larger plates and smaller separations yield larger capacitance. For a **cylindrical capacitor** with inner radius a, outer radius b, and length L, the capacitance is C = 2 pi epsilon_0 L / ln(b/a). For a **spherical capacitor** with inner radius a and outer radius b, the capacitance is C = 4 pi epsilon_0 ab / (b - a). In the limiting case where the outer radius approaches infinity, giving an isolated sphere, the capacitance becomes C = 4 pi epsilon_0 a. ### II. Capacitors in Series and Parallel In a **parallel combination**, all capacitors share the same voltage V. The total charge is the sum of the individual charges (Q_total = Q_1 + Q_2 + Q_3 + ...), and the equivalent capacitance is C_eq = C_1 + C_2 + C_3 + ... A parallel combination always increases the total capacitance. In a **series combination**, all capacitors carry the same charge Q. The total voltage is the sum of the individual voltages (V_total = V_1 + V_2 + V_3 + ...), and the equivalent capacitance satisfies 1/C_eq = 1/C_1 + 1/C_2 + 1/C_3 + ... A series combination always decreases the total capacitance. For two capacitors in series, the equivalent capacitance simplifies to C_eq = C_1 C_2 / (C_1 + C_2). It is worth noting that the series and parallel rules for capacitors are opposite to those for resistors: resistors in series add directly, while capacitors in series add reciprocally, and vice versa for parallel combinations. Two circuit diagrams side by side. Left panel (Parallel): Three capacitors C_1, C_2, C_3 connected in parallel between two nodes, with the same voltage V across each. Below, the equivalent single capacitor C_eq = C_1 + C_2 + C_3 is shown. Right panel (Series): Three capacitors C_1, C_2, C_3 connected end-to-end in series, with charge Q labeled on each. Below, the equivalent capacitor with 1/C_eq = 1/C_1 + 1/C_2 + 1/C_3 is shown. Voltage drops V_1, V_2, V_3 are indicated across each series capacitor. ### III. Energy Stored in a Capacitor Work must be done to charge a capacitor, and this energy is stored in the electric field between the plates. The stored energy can be expressed in three equivalent forms: U = (1/2)QV = (1/2)CV^2 = Q^2 / (2C). The most convenient form depends on which quantities are held constant in a given problem. The **energy density**, or energy per unit volume stored in an electric field, is u = (1/2) epsilon_0 E^2. This is a general result that applies to any electric field, not just the field inside a capacitor. For a parallel plate capacitor, the total energy can be verified as U = u x volume = (1/2) epsilon_0 E^2 (Ad). The ability of capacitors to store and rapidly release energy has many practical applications. Camera flashes use capacitors for rapid energy release. Defibrillators store energy in a capacitor and deliver a controlled pulse to the heart. Power conditioning circuits use capacitors to smooth out voltage fluctuations. ### IV. Dielectrics A **dielectric** is an insulating material placed between the plates of a capacitor. Its effect is to increase the capacitance by a factor kappa (the dielectric constant): C = kappa C_0 = kappa epsilon_0 A / d. The dielectric constant satisfies kappa >= 1 for all materials, with kappa = 1 for vacuum. Common values include air (1.00059), paper (3.7), glass (4-10), water (80), and barium titanate (approximately 1200). The physical mechanism behind this effect involves molecular polarization. The external electric field partially aligns the molecular dipoles in the dielectric material. These aligned dipoles create an internal field (E_induced) that opposes the external field, reducing the net field inside the dielectric to E = E_0 / kappa. When the charge on the plates is held fixed, the voltage decreases, and since C = Q/V, the capacitance increases. When the voltage is held fixed (battery stays connected), additional charge flows onto the plates to maintain the voltage, again increasing C. The **permittivity** of a dielectric is defined as epsilon = kappa epsilon_0, and it replaces epsilon_0 in all capacitance formulas when a dielectric is present. ### V. Dielectrics: Energy and Breakdown The presence of a dielectric affects the stored energy differently depending on what is held constant. For constant charge, the energy decreases: U = U_0 / kappa. The energy reduction means the dielectric is pulled into the gap, a real and measurable force. For constant voltage, the energy increases: U = kappa U_0, because the battery does extra work to push additional charge onto the plates. The energy density in a dielectric is u = (1/2) kappa epsilon_0 E^2 = (1/2) epsilon E^2. **Dielectric breakdown** occurs when the electric field exceeds a critical value called the dielectric strength, at which point the insulator becomes conducting. For air, the dielectric strength is approximately 3 x 10^6 V/m. This sets a maximum voltage for any given capacitor design. Lightning is a dramatic example of dielectric breakdown of air. A parallel plate capacitor shown in two states. Left: Without dielectric — uniform electric field E_0 between plates with charge +Q and -Q, voltage V_0 across the plates. Right: With a dielectric slab (shaded, labeled with kappa) inserted between the plates — the field inside the dielectric is reduced to E_0/kappa, induced charges (smaller + and - symbols) appear on the dielectric surfaces opposite to the plate charges, and the voltage is reduced to V_0/kappa. Arrows show the alignment of molecular dipoles within the dielectric slab. ### VI. Problem-Solving Strategies for Capacitor Circuits Complex capacitor circuits are best approached systematically. First, identify the innermost series and parallel groups. Reduce each group to a single equivalent capacitor, and continue reducing until the entire network is replaced by a single equivalent capacitance. Then work backwards to find the charge Q and voltage V on each individual capacitor. The key relationships to track are that capacitors in series share the same charge but have different voltages, while capacitors in parallel share the same voltage but store different charges. Common pitfalls include confusing the series and parallel rules with those for resistors, forgetting that inserting a dielectric with the battery connected produces different results than inserting it with the battery disconnected, and neglecting to account for the energy changes when dielectrics are inserted or removed.