Lecture on Capacitors

Introduction

• Subject: Capacitors (Physics)
• Importance: High scoring in JEE Mains and Advanced exams
• Focus: Concepts and questions asked in JEE Mains and Advanced
• Sections to cover: Basic concepts, dielectrics, RC circuits

Basic Concepts

Capacitors Overview

• A capacitor is a device used to store energy in the form of an electric field.
• It consists of two conductors separated by an insulator (dielectric).
• Electric field ( E) flows from positive to negative plate.
• Examples: Parallel plate, spherical, and cylindrical capacitors.

Key Terms

• Charge (Q): Stored on the positive plate of the capacitor.
• Potential Difference (V): Difference between the potentials on the two plates.
• Capacitance (C): A measure of a capacitor's ability to store charge, given by the formula C = Q/V.
• Capacitance unit: Farad (F).

Capacitance Dependence

• Depends on the shape and size of conductors and the medium between them.
• Independent of the charge and potential difference.

Types of Capacitors

Parallel Plate Capacitor

• Most common type, consists of two parallel plates.
• Capacitance formula: C = ε₀A/d, where ε₀ is the permittivity of free space, A is the area of the plates, and d is the distance between them.

Spherical Capacitor

• Consists of concentric spherical conductors.
• Capacitance formula: C = 4πε₀(R1R2)/(R2 - R1), where R1 and R2 are the radii of the inner and outer spheres respectively.

Cylindrical Capacitor

• Consists of coaxial cylinders.
• Capacitance formula: C = 2πε₀L/ln(r2/r1), where L is the length of the cylinders, r1 and r2 are the radii of the inner and outer cylinders respectively.

Capacitance with Dielectrics

• Dielectric constant (K): A measure of a material's ability to increase capacitance, where K > 1.
• Capacitance with dielectric: C = KC0, where C0 is the original capacitance.
• Partial Dielectric: If only a part of the gap is filled, capacitance is given by: C = ε₀A/(d - t + t/K), where t is the thickness of the dielectric.

Energy Stored in Capacitors

• Energy (U) stored in a capacitor: U = 1/2 CV².
• Other forms: U = Q²/(2C) and U = 1/2 QV.

Charging and Discharging of Capacitors

• Charging: When connected to a battery, the capacitor's potential difference increases until it matches the battery's voltage.
• Discharging: When disconnected from the battery, the capacitor releases its stored energy.

Time Dependence in RC Circuits

• An RC circuit includes a resistor (R) and a capacitor (C).
• During charging: Q(t) = Q0(1 - e^(-t/RC)), where Q0 is the maximum charge and t is time.
• During discharging: Q(t) = Q0e^(-t/RC).

Important Points

• In RC circuits, the capacitor initially behaves like a short circuit (t = 0) and eventually behaves like an open circuit (t → ∞).
• Practical problems involve calculating voltage, charge, and energy at different times.

Problem Solving

• Use concepts to solve problems involving capacitors in series and parallel.
• Apply formulas to find unknown quantities like charge, voltage, capacitance, and energy.

Example Problems

1. Series and Parallel Capacitors:

   • Capacitors in series: 1/C_eq = 1/C1 + 1/C2 + ...
   • Capacitors in parallel: C_eq = C1 + C2 + ...

2. RC Circuit: Calculate charge or voltage at a given time using exponential decay formulas.

3. Dielectric Insertion: Calculate new capacitance and energy when a dielectric is inserted.

Summary

Capacitors are fundamental components in electrical circuits. Understanding their behavior in different configurations, including the effects of dielectrics and time-dependent behavior in RC circuits, is crucial for solving physics problems effectively.