Electrostatics Study Notes

University of the Witwatersrand Department of Physics - Physics I PQR (PHYSI40 lt4t lr42) Notes

5.1: Electrostatics

5.1.1: Introduction

  • Electric Charge

    • Experiments with an ebonite rod rubbed with fur and a glass rod rubbed with silk reveal two types of electric charge: positive and negative.

    • Key observations:

    • Like charges repel one another.

    • Unlike charges attract one another.

    • Ebonite (negative charge) and fur (positive charge) when in contact:

    • Electrons transfer from the fur to the ebonite, making the ebonite negatively charged and the fur positively charged.

5.1.2: Conductors and Insulators

  • Conductors:

    • Materials that allow charges to move readily, e.g., metals.

    • Exhibit electrical conductivity.

  • Insulators:

    • Materials that do not allow charges to move readily, e.g., glass, rubber, most plastics.

    • Exhibit electrical insulation characteristics (dielectrics).

5.1.3: Charging by Induction

  • Charging a body without direct contact.

  • Process:

    1. Start with an uncharged conducting body insulated from its surroundings.

    2. If a positive charge induces the conductor, it becomes negatively charged.

    3. Negative charges on the inducing rod repel electrons in the conductor, causing negative charges (electrons) to move away to the earth.

    4. Upon removing the earth connection, the induced negative charges remain.

    5. When the inducing charge is removed, the distribution of charges in the conductor redistributes.

5.1.4: The Unit of Charge

  • Charge is quantized; smallest discrete charge is that of an electron (; e = -1.60 \times 10^{-19} C) or a proton (+e).

  • SI Unit: Coulomb (C).

    • Defined in terms of current (Ampere) as follows:
      Q=IimestQ = I imes t
      where:

    • $Q$ = charge (in C)

    • $I$ = current (in A)

    • $t$ = time (in s)

5.1.5: Coulomb's Law

  • Describes the force between two point charges:
    F∝Q<em>1Q</em>2r2F \propto \frac{Q<em>1 Q</em>2}{r^2}

  • In SI units: F=14πε<em>0Q</em>1Q2r2F = \frac{1}{4\pi \varepsilon<em>0} \frac{Q</em>1 Q_2}{r^2} where:

    • $\varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 /\text{ N*m}^2$ is the permittivity of free space.

  • This relationship holds for point charges in vacuum.

  • For different media, use:
    F=14πεQ<em>1Q</em>2r2F = \frac{1}{4\pi \varepsilon} \frac{Q<em>1 Q</em>2}{r^2}
    where $\varepsilon$ is the permittivity of the medium.

5.1.6: Electric Field

  • A charge $q2$ in the neighborhood of another charge $Q1$ experiences a force due to the electric field created by $Q_1$.

  • Electric Field Definition:

    • The electric field $E$ at a point is defined as the force $F$ experienced per unit charge $q2$: E=Fq</em>2E = \frac{F}{q</em>2}

    • Directed in the same direction as the force on a positive charge.

  • SI Unit of Electric Field: N/C (Newtons per Coulomb) or V/m (Volts per meter).

  • Calculating the Electric Field: E=Fq<em>2=14πε</em>0Qr2E = \frac{F}{q<em>2} = \frac{1}{4\pi \varepsilon</em>0} \frac{Q}{r^2}

    • Where $E$ is directed away from $Q$ if $Q$ is positive, and toward $Q$ if $Q$ is negative.

5.1.6.1: Principle of Superposition

  • The resultant electric field due to multiple point charges is the vector sum of the electric fields due to each individual charge.

5.1.6.2: Lines of Force

  • Imaginary lines represent the direction of the electric field.

  • A positive charge placed on a line will move along it.

  • Lines cannot cross one another and must originate from positive charges and end on negative charges.

  • The density of lines indicates the strength of the electric field.

5.1.7: Electrostatic Flux

  • Definition: The number of electric field lines passing through a given area is referred to as the electrostatic flux, denoted as $\Phi_E$.

  • Calculating Flux:
    ΦE=E⋅A\Phi_E = E \cdot A
    where $A$ is the area and $E$ is the electric field component perpendicular to A.

5.1.8: Gauss's Law

  • Used for calculating the electric field due to symmetrical charge distributions.

  • For a point charge $q$:

    • Consider an imaginary sphere centered on the charge:
      Φ<em>E=Q</em>encε0\Phi<em>E = \frac{Q</em>{enc}}{\varepsilon_0}

    • Where $Q_{enc}$ is the total charge enclosed by the Gaussian surface.

5.1.8.1: Applications of Gauss's Law
5.1.8.1.1: Electric Field Due to a Charged Conducting Sphere

  • Apply Gauss's law to calculate the field around a charged sphere using a spherical Gaussian surface.

    • The electric field outside the sphere is given by:
      E=14πε0Qr2E = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r^2}

5.1.8.1.2: Electric Field Due to a Plane Conducting Surface

  • The surface charge density is denoted by $\sigma$ (C/m²).

    • The electric field outside the plane is:
      E=σ2ε0E = \frac{\sigma}{2\varepsilon_0}

5.1.9: Electric Potential

  • Definition: Electric potential (V) at a point is defined as the work done per unit charge in bringing a charge from infinity to that point:
    V=WqV = \frac{W}{q}

  • For a point charge:
    V=q4πε0rV = \frac{q}{4\pi \varepsilon_0 r}

  • The SI unit of electric potential is the volt (V).

5.1.10: Electric Field Between Two Parallel Plates

  • Two oppositely charged parallel plates create an electric field $E$ between them.

    • The electric field can be calculated as:
      E=VdE = \frac{V}{d}
      where d is the distance between the plates and $V$ is the potential difference.

5.1.11: Equipotentials

  • Equipotential surfaces are perpendicular to electric field lines.

  • No work is required to move a charge along an equipotential surface.

5.1.12: Electric Dipoles

  • Definition: A pair of equal and opposite charges form an electric dipole.

    • Example: Water molecule.

  • Dipole Moment (p): Defined as:
    p=qimesdp = q imes d
    where q is the charge and d is the separation distance.

5.1.13: The Electron Volt

  • Unit of energy equivalent to the work done when an electron is moved through a potential difference of 1 V.

  • 1eV=1.60imes10−19J1 eV = 1.60 imes 10^{-19} J.

5.1.14: Capacitance

  • Definition: A capacitor is a device that stores charge and energy. Commonly consists of two parallel plates.

  • Capacitance (C): Defined as: C=QVC = \frac{Q}{V} where Q is charge and V is the voltage across the plates.

    • SI Unit: Farad (F), where $1 F = 1 C/V$.

5.1.14.1.1: Electric Field of Parallel Plates

  • For parallel plates separated by distance d, the electric field is:
    E=VdE = \frac{V}{d} .

5.1.14.2: Capacitors in Series and Parallel

  • Series:

    • Total capacitance:
      1C<em>total=1C</em>1+1C<em>2+…+1C</em>n\frac{1}{C<em>{total}} = \frac{1}{C</em>1} + \frac{1}{C<em>2} + … + \frac{1}{C</em>n}

  • Parallel:

  • Total:
    C<em>total=C</em>1+C<em>2+…+C</em>nC<em>{total} = C</em>1 + C<em>2 + … + C</em>n

5.1.14.3: Energy Stored in a Capacitor

  • Energy (W) stored is calculated as:
    W=12CV2W = \frac{1}{2} C V^2

  • Represented graphically with work done due to the charge moving from one plate to another under potential difference.

5.1.14.4: Dielectrics

  • Importance of dielectrics in capacitors:

    1. Insulate plates.

    2. Maintain constant plate separation.

    3. Increase capacitance.

    4. Allow higher voltage operation.

  • The dielectric reduces the electric field due to charge polarization, counteracting the field between the plates:
    C=KC<em>0C = K C<em>0 where $C0$ is the capacitance with vacuum and $K$ is the dielectric constant.

5.1.15: Transients

  • In circuits with capacitance, when a switch is closed, currents and voltages do not change instantaneously; they undergo transient behavior.

    • Described by exponential equations related to time constant $\tau = RC$ (for a resistor-capacitor circuit).