Capacitance and Dielectrics

Introduction

This study guide covers the fundamental concepts of capacitance and dielectrics, including the behavior of capacitors in various circuit configurations, the calculation of stored energy, and the role of dielectric materials. These topics are essential for understanding electric circuits and electrostatics in college-level physics.

Charge and Potential of Different Shapes of Capacitors

Capacitor Basics

• Capacitor: A device that stores electric potential energy by maintaining separated positive and negative charges. Work is required to separate these charges.

• Capacitance (C): The ability of a capacitor to store charge per unit potential difference.

• Unit of Capacitance: The farad (F), where .

General Formula for Capacitance

• For any capacitor: where

• Capacitance depends on geometric factors and material properties.

Capacitance of Common Geometries

• Parallel Plate Capacitor: where is plate area, is separation, is vacuum permittivity.

• Spherical Capacitor:

• Cylindrical Capacitor: where is length, and are radii.

Capacitors in Circuits

Symbols for Capacitors

• Standard symbol: two parallel lines (a)

• Electrolytic capacitor: one straight, one curved line (b)

• Variable capacitor: parallel lines with a diagonal arrow (c)

Capacitors in Parallel

When capacitors are connected in parallel, the total capacitance increases.

• Key Properties:

  • Voltage across each capacitor is the same.

  • Total charge is the sum of individual charges.

• Formulas: General case:

Capacitors in Series

When capacitors are connected in series, the total capacitance decreases.

• Key Properties:

  • Charge on each capacitor is the same.

  • Total voltage is the sum of individual voltages.

• Formulas: General case:

Example: Equivalent Capacitance Calculation

• Given:

• and in parallel:

• and in series:

Example: Charge Distribution

• Given ,

• Voltage across :

• Voltage across :

• Charge on and :

Energy Stored in Capacitors

Energy Storage Principle

• Work is required to move charge onto the plates of a capacitor.

• Energy stored (U):

Energy Density in Electric Field

• For a parallel plate capacitor, energy density is: where is the electric field between plates.

Dielectrics

Dielectric Materials

• Dielectric: An insulating material placed between capacitor plates to increase capacitance.

• Dielectric Constant (): Ratio of capacitance with dielectric to that without: where is the capacitance without dielectric.

• Polar Dielectrics: Molecules have permanent dipole moments; align with external field, creating an opposing internal field.

• Non-polar Dielectrics: Molecules acquire dipole moments in an external field, also opposing the field.

Effect of Dielectric on Electric Field

• Electric field inside dielectric is reduced by factor .

• Permittivity of dielectric:

• Inside dielectric:

Dielectric Strength

• Dielectric Strength: Maximum electric field a dielectric can withstand before breakdown.

• Dielectric breakdown occurs when the field is strong enough to ionize the material.

Example: Capacitance with Dielectric

• Two aluminum sheets, area , separated by waxed paper (, ).

• Capacitance without dielectric:

• Capacitance with dielectric:

Capacitor Behavior with Dielectric and Battery

Battery Connected

• Inserting dielectric with battery connected: remains constant, increases, increases.

Battery Disconnected

• Inserting dielectric after disconnecting battery: remains constant, decreases, increases.

Summary Table: Capacitor Combinations

Key Equations Summary

• Parallel:

• Series:

• Energy stored:

• Energy density:

• Capacitance with dielectric:

Additional info: The notes reference Gauss' Law in dielectrics, but do not provide details. In academic context, Gauss' Law is used to relate the electric displacement field to free charge in the presence of dielectrics: , where and is polarization.