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 energy stored, and the role of dielectrics in modifying capacitance. These topics are essential for understanding electric circuits and electrostatics in college-level physics.

Charge and Potential of Different Shapes of Capacitors

Capacitors are devices that store electric potential energy by maintaining separated positive and negative charges. The relationship between charge (Q) and potential difference (V) depends on the geometry of the capacitor.

• Parallel Plate Capacitor: Where is the vacuum permittivity, is the plate area, and is the separation.

• Spherical Capacitor: Where and are the radii of the inner and outer spheres.

• Cylindrical Capacitor: Where is the length, and are the radii.

Capacitance is defined as:

• Where

• The unit of capacitance is the farad (F):

Symbols for Capacitors in Circuit Diagrams

Capacitors are represented in circuit diagrams by specific symbols:

• (a) Standard capacitor: two parallel lines

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

• (c) Variable capacitor: parallel lines with an arrow

Capacitors in Parallel

When capacitors are connected in parallel, the total charge is the sum of the charges on each capacitor, and the voltage across each is the same.

• For n capacitors:

Capacitors in Series

For capacitors in series, the charge on each is the same, but the total voltage is the sum of the voltages across each capacitor.

• For n capacitors:

Example: Equivalent Capacitance Calculation

Given three capacitors arranged as shown:

• and are in parallel:

• and are in series:

Example: Charge and Voltage Distribution

• Given ,

• Voltage across :

• Voltage across :

• Charge on and :

Energy Stored in Capacitors

The energy stored in a capacitor is the work required to separate the charges:

• Total work:

• For a capacitor:

Energy Density in Electric Field

For a parallel plate capacitor, the energy density (u) in the electric field is:

• Where is the electric field between the plates.

Dielectrics

Dielectrics are insulating materials placed between capacitor plates to increase capacitance. They affect the electric field and energy storage properties.

• Dielectric constant (κ): 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, leading to similar effects as polar dielectrics.

• Permittivity:

• Dielectric strength: Maximum electric field before breakdown occurs.

Effects of Dielectrics in Circuits

• With battery connected: constant, increases, increases.

• Battery disconnected: constant, decreases, increases.

Example: Capacitance with Dielectric

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

• Calculation:

Summary Table: Capacitor Combinations

Summary Table: Key Formulas

Additional info:

• Gauss' Law in dielectrics modifies the electric field calculation by accounting for the permittivity of the dielectric material.

• Dielectric breakdown is a critical limit for capacitor operation, especially in high-voltage applications.