04 Albach Kapitel 2.1 - 2.9

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Schlüsselkonzepte:

• Der elektrische Strom (Kap. 2.1)
• Spezifische Leitfähigkeit (inkl. Temperaturabhängigkeit) (Kap. 2.5)
• Das Ohm‘sche Gesetz (Kap. 2.6)
• Energie und Leistung (Kap. 2.9)

Lernziele:

• Die Stromdichteverteilung in einfachen Anordnungen zu berechnen 
• Den ohmschen Widerstand von einfachen Leiteranordnungen zu berechnen 
• Die Temperaturabhängigkeit der ohmschen Widerstände anzugeben  
• Das Ohm‘sche Gesetz in differentieller und integraler Form anzuwenden
• Das Verhalten der Stromdichte an Materialsprungstellen mit unterschiedlichen Leitfähigkeiten zu bestimmen 
• Energie und Leistung im stationären Strömungsfeld zu berechnen

2.1 The Electric Current ($I$) ⚡

Electric current is the directed movement of electric charge carriers, such as electrons in a wire or ions in a fluid1111111.

• Cause: A current flows when a potential difference (voltage) is applied across a conductive material, creating an electric field $\vec{E}$ that exerts a force on the charge carriers 2.

• Technical Direction: By convention, the “technical” direction of current is defined as the direction positive charges would flow, i.e., from a higher potential to a lower potential3.

• Electron Flow: Since electrons are negatively charged, their physical movement (drift) is opposite to the conventional technical current direction4444.

• Formula: Current $I$ is a scalar quantity 555representing the total charge $\Delta Q$ that passes through a cross-section per unit time $\Delta t$6666:

  $I=\frac{\Delta Q}{\Delta t}$

  For time-varying currents, this becomes $I(t)=dQ/dt$7.

• Unit: The base SI unit for electric current is the **Ampere (A)**8888. As of 2019, the Ampere is defined by fixing the numerical value of the elementary charge $e$ to $\mathrm{1.602...}\times 10^{-19}$ C (or As)999. One Ampere corresponds to $1/e$ (approx. $6.24\times 10^{18}$) elementary charges passing per second10.

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2.2 The Current Density ($\vec{J}$)

While current $I$ is a scalar, current density $\vec{J}$ is a vector field that describes the direction and density of current flow at a specific point in space 11111111111111.

• Microscopic View: Current density is the product of the volume charge density $\rho$ of the mobile carriers and their average drift velocity $\vec{v}$12:

  $\vec{J}=\rho \vec{v}$

• Relationship to Current: The total current $I$ flowing through a surface $A$ is the flux (surface integral) of the current density vector $\vec{J}$ through that surface13. This is calculated using a scalar product14:

  $I={\iint }_A\vec{J}\cdot d\vec{A}\text{[cite: 7476]}$

  For a uniform current density perpendicular to a flat area $A$, this simplifies to $J=I/A$.

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2.3 The Stationary Current Field

A current field is “stationary” (or DC) if the current density $\vec{J}$ is constant in time at every point15.

• Continuity Equation: In a stationary field, charge cannot “pile up” or be depleted from any point. This means the total current flux out of any closed surface $A$ must be zero16:

  ${\oint }_A\vec{J}\cdot d\vec{A}=0$

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2.4 & 2.5 Specific Conductivity ($\kappa$) and Resistivity (${\rho }_R$)

These are material properties that link the electric field $\vec{E}$ to the resulting current density $\vec{J}$.

• Drift Velocity: In a conductor, free electrons move randomly17171717. An applied $\vec{E}$-field causes them to “drift” with an average velocity ${\vec{v}}_e$. This drift velocity is proportional to the field: ${\vec{v}}_e=-{\mu }_e\vec{E}$, where ${\mu }_e$ is the electron mobility      .

• Specific Conductivity ($\kappa$): Combining the microscopic view $\vec{J}=\rho {\vec{v}}_e=(-ne){\vec{v}}_e$ (where $n$ is the density of free electrons) with the drift velocity gives:

  $\vec{J}=(-ne)(-{\mu }_e\vec{E})=(ne{\mu }_e)\vec{E}\text{[cite: 5229-5230, 7298]}$

  The material-dependent term $\kappa =ne{\mu }_e$ is the specific conductivity 20202020. Its unit is $1/(\Omega \text{m})$21. Good conductors like silver and copper have high $\kappa$ values22222222.

• Specific Resistivity (${\rho }_R$): This is the reciprocal of conductivity: ${\rho }_R=1/\kappa$23.

• Temperature Dependence: For most metals, resistivity increases with temperature24. This is because higher thermal vibrations of the atoms cause more frequent collisions, reducing electron mobility ${\mu }_e$25. The relationship is approximately linear 26:

  ${\rho }_R(T)={\rho }_{R,20^{\circ }C}\cdot [1+\alpha (T-20^{\circ }C)]\text{[cite: 7333]}$

  Here, $\alpha$ is the temperature coefficient27272727.

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2.6 Ohm’s Law

Ohm’s Law describes the relationship between voltage, current, and resistance. It has two forms:

• Differential Form: This is the fundamental local relationship:

  $\vec{J}=\kappa \vec{E}\text{[cite: 5246, 7534]}$

  This implies that, unlike in electrostatics, a non-zero $\vec{E}$-field must exist inside a conductor that is carrying a current28282828.

• Integral Form (Resistance $R$): For a component, the ohmic resistance $R$ is defined as the ratio of the total voltage $U$ across it to the total current $I$ through it29:

  $R=\frac{U}{I}=\frac{\int \vec{E}\cdot d\vec{s}}{\iint \kappa \vec{E}\cdot d\vec{A}}\text{[cite: 7548]}$

  For a simple wire with length $l$ and uniform cross-sectional area $A$, this simplifies to:

  $R=\frac{{\rho }_{R}l}{A}=\frac{l}{\kappa A}\text{[cite: 5257, 7375]}$

  This shows resistance increases with length $l$ 30and decreases with area $A$ (which is proportional to the diameter squared, $A\sim d^2$)31. The familiar integral form is $U=R\cdot I$32.

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2.7 Practical Resistors

Resistors are components built to provide a specific resistance3333.

• Fixed Resistors: Their values are standardized in E-series (e.g., E6, E12, E24), which define the set of values and their tolerance (e.g., ±20%, ±10%, ±5%) 3434343434343434. Values are often marked with a color code 35353535.

• Types: Include layer resistors and wire-wound resistors3636363636363636. Wire-wound resistors can have “bifilar” or “low-inductance” windings to minimize parasitic magnetic field effects 37373737.

• Variable Resistors: These include potentiometers (adjustable) 38383838, varistors (voltage-dependent), LDRs (light-dependent), and thermistors (NTC/PTC, temperature-dependent) 39393939.

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2.8 Field Behavior at Interfaces

When current flows across a boundary between two materials with different conductivities (${\kappa }_1,{\kappa }_2$), the field vectors must obey boundary conditions40404040.

• Normal Component $J_n$: The current density component perpendicular to the boundary is continuous:

  $J_{n1}=J_{n2}\text{[cite: 5237, 7718]}$

• Tangential Component $E_t$: The electric field component parallel to the boundary is continuous:

  $E_{t1}=E_{t2}\text{[cite: 5239, 7726]}$

• Consequences: Using $\vec{J}=\kappa \vec{E}$, this means the other components jump:

  • $J_{t2}=({\kappa }_2/{\kappa }_1)J_{t1}$ 41

  • $E_{n2}=({\kappa }_1/{\kappa }_2)E_{n1}$

• Extreme Cases:

  • Conductor to Insulator (${\kappa }_2=0$): $J_{n2}=0$, so $J_{n1}=0$. Current must flow parallel to the surface42424242.

  • Conductor to Perfect Conductor (${\kappa }_2\to \infty$): $E_{t2}=0$, so $E_{t1}=0$. The $\vec{E}$ and $\vec{J}$ fields must enter a perfect conductor perpendicularly 43434343.

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2.9 Energy and Power 🔥

• Electric Power ($P$): The rate at which energy is converted. It is defined as:

  $P=U\cdot I\text{[cite: 5268, 7784]}$

  Its unit is the **Watt (W)**444444.

• Electric Energy ($W_e$): The total energy converted over a time $\Delta t$.

  $W_e=P\cdot \Delta t\text{[cite: 5268, 7786]}$

  Its unit is the Joule (J) or Watt-second (Ws). A common billing unit is the kilowatt-hour (kWh) 454545.

• Joule Heating: In an ohmic resistor, this power is dissipated as heat46. The power can be expressed as:

  $P=I^{2}R=\frac{U^2}{R}\text{[cite: 5268, 7876]}$

• Power Density ($p_v$): The dissipated power per unit volume (W/m³). This is the differential form of Joule’s Law:

  $p_v=\vec{E}\cdot \vec{J}=\kappa E^2=\frac{J^2}{\kappa }\text{[cite: 5269]}$