CHPTR 24 Notes: Capacitance & Dielectric

Capacitors

  • Definition:

    • A capacitor consists of two conductors that are isolated from each other and their surroundings.

    • When a capacitor is charged (by connecting it to a battery), equal magnitudes of opposite charges accumulate on each conductor.

  • Equi-Potential Surfaces:

    • The plates of the capacitor, being conductors, are equi-potential surfaces, meaning that the potential is the same at all points on each plate.

Common Uses of Capacitors

  • Applications:

    • Tuning receivers.

    • Filtering in power supplies.

    • Storing energy and charge.

  • Energy Storage:

    • Unlike batteries, capacitors can discharge energy rapidly.

    • Immediate applications include:

    • Lighting flashbulbs.

    • Powering defibrillators.

Capacitive Touch Sensors

  • Operating Principle:

    • Utilizes a parallel plate capacitor model, where:

    • One plate acts as the sensor’s electrode.

    • The other plate is the human finger or another grounded target.

    • Changes in the gap between the plates alter capacitance, which is converted into a voltage signal for touch detection.

  • Electric Field Detection:

    • An electric field is generated above the touchscreen. The human body disturbs this field upon nearing the screen.

    • The disturbance results in a change in the local electrostatic field, indicating a touch event.

  • Visual Representation of Electric Fields:

    • Picture electric field lines extending into the air from the sensor. Moving a finger near bends these lines, akin to a magnet affecting iron filings.

  • Application of Capacitive Principles:

    • Capacitive touch screens provide quick and precise touch detection through electric field changes and capacitive coupling.

    • Capable of detecting not just direct touch but also proximity, broadening their use across various applications.

Charging Capacitors

  • Process:

    • When connected to a battery, an electric field arises in the connecting wires, facilitating charge flow onto the capacitor plates.

    • Charging continues until the potential difference across the plates matches the terminal voltage of the battery.

    • At full charge, the electric field in the wires collapses, leaving a field present between the plates.

Capacitance

  • Definition:

    • Capacitance ($C$) is defined as the ratio of charge ($Q$) on either conductor to the potential difference ($V$) between them.

    • Formula:
      C = rac{Q}{V}

  • Units:

    • The unit of capacitance is the Farad (F). A Farad is a large capacity; typically, in practical applications, capacitances are expressed in microfarads (µF) or picofarads (pF).

Parallel-Plate Capacitors

  • Characteristics:

    • The capacitance of a parallel-plate capacitor is proportional to the area ($A$) of the plates and inversely proportional to the separation distance ($d$) between them.

    • Formula:
      C = rac{ ext{ε}_0 A}{d}

Spherical Capacitors

  • Capacitance Formula:

    • For spherical capacitors, the capacitance ($C$) is given by:
      C = rac{4 ext{π ε}_0}{ rac{1}{b} - rac{1}{a}} where $a$ and $b$ are the inner and outer radii.

Cylindrical Capacitors

  • Capacitance per Unit Length:

    • The formula is given by:
      C = rac{2 ext{π ε}_0}{ ext{ln}( rac{b}{a})}

    • Here, $a$ is the inner radius and $b$ is the outer radius of the cylindrical capacitor.

Circuit Symbols

  • Basic circuit symbols for various electrical components, including capacitors, are presented in a diagram in the course material.

Combination of Capacitors

Parallel Combination

  • Voltage:

    • Each capacitor in parallel has the same voltage across it.

  • Equivalence Formula:

    • The equivalent capacitance ($C{eq}$) is calculated as: C{eq} = C1 + C2 + C_3 + …

Series Combination

  • Charge Characteristics:

    • For capacitors connected in series, the charge ($Q$) on each capacitor is identical.

  • Equivalence Relation:

    • Given their individual voltages ($V1$, $V2$, …), the equivalent capacitance is found using:
      rac{1}{C{eq}} = rac{1}{C1} + rac{1}{C_2} + …

Equivalent Capacitances

  • To find the total equivalent capacitance of several capacitors:

    1. Calculate equivalent capacitances of capacitors in parallel first.

    2. Then add them in series if applicable.

Energy Stored in a Capacitor

  • Work Done:

    • An external agent, like a battery, must perform work to charge a capacitor, thus storing energy.

  • Energy Calculation:

    • The energy ($U$) stored in a capacitor is represented as:
      U = rac{1}{2} C V^2 = rac{1}{2} Q V

    • The area under the voltage versus charge ($V$ vs. $Q$) graph corresponds to the energy.

Energy in Electric Fields

  • Energy Storage Context:

    • The energy can be interpreted as stored within the electric field of the capacitor.

  • Energy Density Definition:

    • The energy density () in any electric field relates to the square of the electric field intensity:
      u = rac{1}{2} rac{E^2}{ ext{ε}_0}

    • Integrational forms regarding the volume and area relation are also described.

Capacitors with Dielectrics

  • Dielectric Definition:

    • A dielectric is a non-conducting medium, such as rubber or glass, that enhances capacitance when present.

  • Capacitance Increase:

    • The capacitance increases by a factor known as the dielectric constant ($ ext{kappa}$).

  • Formula with Dielectric:

    • For a capacitor exposed to a dielectric:
      C = ext{kappa} imes C0 where $C0$ is the capacitance without the dielectric.

  • Dielectric Constant Table (approx. values at 20°C):

    • Vacuum: 1

    • Polyvinyl Chloride: 3.18

    • Air (1 atm): 1.00059

    • Plexiglass: 3.40

    • Water: 80.4

    • Strontium Titanate: 310

Keyboards Using Capacitors

  • Some keyboards employ capacitors positioned beneath the keys. As the spacing between plates diminishes, the resulting increased capacitance can be detected by circuits.

Dielectrics

  • Modification of Electrostatic Equations:

    • The behavior of dielectrics is represented in electric equations by modifying the permittivity constant:
      ext{E} = rac{1}{4 ext{π} ext{ε}_0 ext{kappa}}

  • This reflects how dielectrics affect electric fields in their presence.

Atomic Description of Dielectrics

  • Molecular Properties:

    • Dielectrics consist of molecules that either have permanent dipoles or can develop dipoles under external electric fields.

    • The resultant effect in an electric field aligns the dipole moments with the field, causing a reduced net electric field inside the dielectric material.

Dielectric Strength

  • Definition:

    • The dielectric strength is the maximum electric field strength that a dielectric can withstand before breakdown occurs.

  • Advantages of Dielectrics:

    • Increased capacitance.

    • Enhanced maximum operating voltage.

    • Provides mechanical support to structures.

Dielectric Breakdown

  • Example of dielectric breakdown in a block of Plexiglass, which shows dramatic effects, like lightning.

  • Strength Value:

    • Plexiglass has a dielectric strength of about 30 imes 10^6 ext{V/m}, which is nearly 10 times the strength of air.

Types of Capacitors

  • Commercial Manufacturing:

    • Capacitors are frequently constructed from metallic foil combined with thin layers of paraffin paper.

    • High-voltage capacitors consist of numerous interwoven metallic plates that are submerged in silicone oil.