Energy Stored in Capacitors

  • Capacitors store energy in the form of electric field
  • Energy stored in a capacitor with capacitance C and voltage V is given by the equation:
    • E = 0.5 * C * V^2
  • The energy stored in a capacitor is dependent on its capacitance and the applied voltage

Field in Dielectrics

  • A dielectric is an insulating material that can be placed between the plates of a capacitor
  • It increases the capacitance and reduces the electric field between the plates
  • The electric field inside a dielectric can be calculated using the equation:
    • E = V / d
  • Where E is the electric field, V is the voltage, and d is the distance between the plates

Gauss’s Law in Dielectrics - Parallel Plate Capacitor

  • Gauss’s Law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface
  • For a parallel plate capacitor with a dielectric material between the plates, the electric field is given by the equation:
    • E = σ / ε
  • Where E is the electric field, σ is the charge density, and ε is the permittivity of the dielectric material

Example: Energy Stored in Capacitors

  • Consider a capacitor with a capacitance of 10 μF and a voltage of 100 V
  • Calculating the energy stored in the capacitor using the formula:
    • E = 0.5 * C * V^2
    • E = 0.5 * 10 * 10^(-6) * (100^2)
    • E = 0.5 * 10^(-4) * 10,000
    • E = 5 joules

Example: Field in Dielectrics

  • Let’s consider a parallel plate capacitor with a dielectric material of thickness 2 cm
  • The voltage across the plates is 50 V
  • Calculating the electric field inside the dielectric using the formula:
    • E = V / d
    • E = 50 / 0.02
    • E = 2500 V/m

Example: Gauss’s Law in Dielectrics - Parallel Plate Capacitor

  • Suppose we have a parallel plate capacitor with a charge density of 5 μC/m^2 and a dielectric material with a permittivity of 8.85 x 10^(-12) F/m
  • Calculating the electric field inside the dielectric using the equation:
    • E = σ / ε
    • E = 5 x 10^(-6) / (8.85 x 10^(-12))
    • E ≈ 5.65 x 10^5 N/C

Capacitance - Definition

  • Capacitance is a measure of a capacitor’s ability to store charge
  • It is defined as the ratio of the charge stored on the capacitor to the voltage across it
  • The formula for capacitance is:
    • C = Q / V
  • Where C is the capacitance, Q is the charge, and V is the voltage

Capacitance - Unit

  • The SI unit of capacitance is the Farad (F)
  • 1 Farad is equal to 1 Coulomb per Volt (1 F = 1 C/V)
  • Capacitances are usually measured in multiples of the Farad, such as microfarads (μF), nanofarads (nF), and picofarads (pF)

Capacitance - Parallel Plate Capacitor

  • The capacitance of a parallel plate capacitor can be calculated using the formula:
    • C = ε * A / d
  • Where C is the capacitance, ε is the permittivity of the material between the plates, A is the area of the plates, and d is the distance between the plates

Capacitance - Series and Parallel Combinations

  • Capacitors can be connected in series or parallel to get the equivalent capacitance
  • In series combination, the reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of the individual capacitances:
    • 1/Ceq = 1/C1 + 1/C2 + 1/C3 + …
  • In parallel combination, the equivalent capacitance is equal to the sum of the individual capacitances:
    • Ceq = C1 + C2 + C3 + …
  1. Capacitors in Series:
  • When capacitors are connected in series, the same charge is stored on each capacitor.
  • The voltage across each capacitor depends on their capacitance.
  • The reciprocal of the equivalent capacitance in a series combination is equal to the sum of the reciprocals of the individual capacitances:
    • 1/Ceq = 1/C1 + 1/C2 + 1/C3 + …
  • For example, if two capacitors of capacitance 2 μF and 4 μF are connected in series, the equivalent capacitance would be 1.33 μF.
  1. Capacitors in Parallel:
  • When capacitors are connected in parallel, the same voltage is applied across each capacitor.
  • The total charge stored in the combination is the sum of the charges stored in each capacitor.
  • The equivalent capacitance in a parallel combination is equal to the sum of the individual capacitances:
    • Ceq = C1 + C2 + C3 + …
  • For example, if two capacitors of capacitance 3 μF and 5 μF are connected in parallel, the equivalent capacitance would be 8 μF.
  1. Capacitors in Series: Voltage Division:
  • In a series combination of capacitors, the voltage across each capacitor depends on their capacitance.
  • The voltage across each capacitor can be calculated using the formula:
    • V1 = V * (C1 / Ceq)
    • V2 = V * (C2 / Ceq)
    • V3 = V * (C3 / Ceq)
  • For example, if two capacitors of capacitance 4 μF and 6 μF are connected in series with a voltage of 100 V, the voltage across the 4 μF capacitor would be 40 V and the voltage across the 6 μF capacitor would be 60 V.
  1. Capacitors in Parallel: Charge Division:
  • In a parallel combination of capacitors, the charge stored on each capacitor depends on their capacitance.
  • The charge stored on each capacitor can be calculated using the formula:
    • Q1 = Q * (C1 / Ceq)
    • Q2 = Q * (C2 / Ceq)
    • Q3 = Q * (C3 / Ceq)
  • For example, if two capacitors of capacitance 5 μF and 7 μF are connected in parallel with a charge of 20 μC, the charge stored on the 5 μF capacitor would be 10 μC and the charge stored on the 7 μF capacitor would be 14 μC.
  1. Dielectric Strength:
  • The dielectric strength is the maximum electric field a dielectric material can withstand before it breaks down and conducts electricity.
  • Dielectric strength is usually measured in volts per meter (V/m).
  • Different dielectric materials have different dielectric strengths.
  • The dielectric strength determines the maximum voltage that can be applied across a dielectric without causing breakdown.
  1. Dielectric Constant:
  • The dielectric constant of a material is a measure of how well it can polarize in an electric field.
  • It is the ratio of the capacitance with the dielectric material to the capacitance without the dielectric material.
  • The dielectric constant is a dimensionless quantity.
  • Different materials have different dielectric constants.
  • Dielectric constant affects the capacitance of a capacitor with a dielectric material.
  1. Dielectric Constant Formula:
  • The dielectric constant (k) can be calculated using the formula:
    • k = C / Co
  • Where k is the dielectric constant, C is the capacitance with the dielectric material, and Co is the capacitance without the dielectric material.
  1. Dielectric Polarization:
  • When a dielectric material is placed in an electric field, the positive and negative charges within the material are displaced.
  • This creates a polarization in the dielectric material.
  • The polarization is the separation of positive and negative charges within a dielectric material due to an external electric field.
  • The polarization increases the electric field inside the dielectric and reduces the overall electric field between the plates of a capacitor.
  1. Dielectric Polarization: Capacitance Increase:
  • The presence of a dielectric material between the plates of a capacitor increases the capacitance.
  • The increase in capacitance is due to the polarization of the dielectric material, which increases the amount of charge stored on the plates for the same applied voltage.
  • The capacitance of a capacitor with a dielectric material is given by the equation:
    • C = k * Co
  • Where C is the capacitance with the dielectric, k is the dielectric constant, and Co is the capacitance without the dielectric.
  1. Dielectric Polarization: Physiological Effects:
  • Dielectric materials are widely used in various applications, including capacitors used in electronic devices.
  • The presence of dielectrics in electronic devices protects users from electric shocks by insulating them from high voltages.
  • Dielectrics also improve energy storage and release in capacitors, leading to efficient energy transfer in circuits.
  • The choice of dielectric material depends on factors such as dielectric constant, dielectric strength, and thermal conductivity.

Resistivity

  • Resistivity is a measure of a material’s opposition to electric current
  • It is denoted by the symbol ρ (rho) and is defined as the resistance of a material with a unit length and unit cross-sectional area
  • The formula for resistivity is:
    • ρ = R * A / L
  • Where ρ is the resistivity, R is the resistance, A is the cross-sectional area, and L is the length of the material

Ohm’s Law

  • Ohm’s Law states that the current flowing through a conductor is directly proportional to the voltage applied across it, and inversely proportional to its resistance
  • The formula for Ohm’s Law is:
    • V = I * R
  • Where V is the voltage, I is the current, and R is the resistance

Electric Power

  • Electric power is the rate at which electric energy is transferred or used
  • The formula for electric power is:
    • P = V * I
  • Where P is the power, V is the voltage, and I is the current

Electric Circuit

  • An electric circuit is a closed loop through which electric current can flow
  • It consists of a source of electrical energy, such as a battery or power supply, and various components such as resistors, capacitors, and inductors
  • Components in a circuit are connected by conductive materials such as wires

Series Circuit

  • In a series circuit, the components are connected one after another, forming a single pathway for the current to flow
  • The current through each component is the same, and the total resistance is the sum of the individual resistances
  • The formula for calculating the total resistance in a series circuit is:
    • R_total = R1 + R2 + R3 + …

Parallel Circuit

  • In a parallel circuit, the components are connected in separate branches, providing multiple pathways for the current to flow
  • The voltage across each component is the same, and the total resistance is calculated differently from a series circuit
  • The formula for calculating the total resistance in a parallel circuit is:
    • 1/R_total = 1/R1 + 1/R2 + 1/R3 + …

Kirchhoff’s Laws

  • Kirchhoff’s Laws are fundamental principles in circuit analysis
  • 1st Law (Kirchhoff’s Current Law): The sum of currents entering a junction in a circuit is equal to the sum of currents leaving the junction
  • 2nd Law (Kirchhoff’s Voltage Law): The sum of voltages around any closed loop in a circuit is equal to zero

RC Circuit

  • An RC circuit is a combination of a resistor (R) and a capacitor (C) connected in either series or parallel
  • The time constant (τ) of an RC circuit is a measure of how quickly the capacitor charges or discharges
  • The time constant can be calculated using the formula:
    • τ = R * C

RC Circuit - Charging

  • When a capacitor is connected in series with a resistor and a voltage is applied, the capacitor charges up
  • The charging process follows an exponential curve, and the voltage across the capacitor can be calculated using the formula:
    • V = V0 * (1 - e^(-t/τ))
  • Where V is the voltage across the capacitor at time t, V0 is the initial voltage, t is the time, and τ is the time constant

RC Circuit - Discharging

  • When a charged capacitor is connected in series with a resistor and the circuit is closed, the capacitor discharges
  • The discharging process also follows an exponential curve, and the voltage across the capacitor can be calculated using the formula:
    • V = V0 * e^(-t/τ)
  • Where V is the voltage across the capacitor at time t, V0 is the initial voltage, t is the time, and τ is the time constant