03 Albach Kapitel 1.14 - 1.21

Material: HS25_NuS1_Kapitel_1_El_Stat_Feld_2von3 1.pdf Experiment: Faraday-Käfig Experiment: Kondensator treibt Motor

Dielectric Polarization (Kap. 1.14)

When an insulating material, or dielectric, is placed in an electric field, its charges are not free to move like in a conductor. Instead, the atoms and molecules polarize.

• Mechanisms: This happens in two main ways:

  1. Displacement Polarization (Verschiebungspolarisation): The atom’s negative electron cloud shifts slightly in the opposite direction of the $\vec{E}$-field, while the positive nucleus shifts with it. This creates a tiny electric dipole.

  2. Orientation Polarization (Orientierungspolarisation): Molecules that are already permanent dipoles (like water) align themselves with the external $\vec{E}$-field.

• Effect: This alignment of dipoles creates polarization charges on the surface of the dielectric. These charges produce an internal electric field that opposes the external field, thus reducing the total $\vec{E}$-field inside the material.

• Permittivity: This material property is described by the relative permittivity (or dielectric constant), ${ϵ}_r$. It’s a unitless value indicating how much the field is weakened. For a vacuum, ${ϵ}_r=1$. For air, it’s very close to 1. For materials like water or ceramics, it can be much higher (${ϵ}_r>1$).

• Flux Density in Dielectrics: The electric flux density $\vec{D}$ is modified to account for the material:

  $\vec{D}=ϵ\vec{E}={ϵ}_r{ϵ}_0\vec{E}$

  Here, $ϵ={ϵ}_r{ϵ}_0$ is the permittivity of the material.

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Field Behavior at Interfaces (Kap. 1.16)

When an electric field crosses the boundary between two different materials (e.g., from vacuum, ${ϵ}_1={ϵ}_0$, to a dielectric, ${ϵ}_2={ϵ}_r{ϵ}_0$), the field vectors must obey specific boundary conditions. Assuming there is no free charge $\sigma$ on the boundary itself:

• Tangential $\vec{E}$-Field ($E_t$): The tangential component of the electric field $\vec{E}$ is continuous (it’s the same on both sides). This is a consequence of the conservative nature of the $\vec{E}$-field ($\oint \vec{E}\cdot d\vec{s}=0$).

• Normal $\vec{D}$-Field ($D_n$): The normal component of the electric flux density $\vec{D}$ is continuous (it’s the same on both sides). This is a consequence of Gauss’s Law ($\oint \vec{D}\cdot d\vec{A}=0$ when no free charge is enclosed).

• Implications: Because $\vec{D}=ϵ\vec{E}$, these two rules mean the other components must jump (be discontinuous):

  • $D_{t1}\ne D_{t2}$ (Tangential $\vec{D}$ jumps)

  • $E_{n1}\ne E_{n2}$ (Normal $\vec{E}$ jumps)

This leads to a “refraction” of the field lines as they cross the boundary.

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Capacitance (Kap. 1.17)

Capacitance ($C$) is the property of a conductor arrangement to store electric charge and energy. It is defined as the ratio of the stored charge $Q$ on one conductor to the voltage $U$ (potential difference) between the conductors.

• Formula: $C=Q/U$

• Unit: The unit is the Farad (F), where $1\text{ Farad}=1\text{ Coulomb}/1\text{ Volt}$ ($1\text{ As/V}$).

• Calculation: Capacitance depends only on the geometry of the conductors and the permittivity ($ϵ$) of the dielectric material between them.

  • Parallel-Plate Capacitor: For two plates of area $A$ separated by a distance $d$:

    $C={ϵ}_r{ϵ}_0\frac{A}{d}$

  • Spherical Capacitor: For concentric spheres with radii $r_a$ (outer) and $r_i$ (inner):

    $C=4\pi {ϵ}_r{ϵ}_0\frac{r_i{r}_a}{r_a-r_i}$

  • Coaxial Cable (Cylindrical Capacitor): The capacitance per unit length ($C/l$) for cylinders with radii $r_a$ (outer) and $r_i$ (inner) is:

    [cite_start]

    $C/l=\frac{2\pi {ϵ}_r{ϵ}_0}{\ln (r_a/r_i)}\text{[cite: 3418]}$

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Capacitor Networks (Kap. 1.18)

Capacitors can be combined in simple networks:

• Parallel Connection:

  • The voltage $U$ across each capacitor is the same.

  • The total charge stored is the sum $Q_{ges}=Q_1+Q_2+...$.

  • The total capacitance is the sum of the individual capacitances:

    $C_{ges}=C_1+C_2+...+C_n$

• Series Connection:

  • The charge $Q$ on each capacitor is the same (due to influence).

  • The total voltage is the sum $U_{ges}=U_1+U_2+...$.

  • The reciprocals of the capacitances add up:

    $\frac{1}{C_{ges}}=\frac{1}{C_1}+\frac{1}{C_2}+...+\frac{1}{C_n}$

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Energy in the Electrostatic Field (Kap. 1.21) ⚡

The work done to charge a capacitor is stored as potential energy $W_e$ in the electric field.

• Energy in a Capacitor: This stored energy can be expressed in three equivalent ways:

  $W_e=\frac{1}{2}C{U}^2=\frac{1}{2}QU=\frac{1}{2}\frac{Q^2}{C}$

• Energy Density: The energy is not located on the plates but is distributed throughout the volume $V$ of the electric field. The energy density $w_e$ (energy per unit volume, in J/m³) at any point is:

  $w_e=\frac{1}{2}\vec{E}\cdot \vec{D}$

• Total Energy: The total energy $W_e$ is the integral of this density over the entire volume where the field exists:

  $W_e={\iiint }_V{w}_{e}dV$