Integrals and Charge Densities

Theoretical Framework of Integrals

• Discretization Symbols:

  • $dx$: Infinitesimal length element along a 1D axis.

  • $dA$: Infinitesimal area element for 2D surfaces.

  • $dV$: Infinitesimal volume element for 3D objects.

  • $dq$: Infinitesimal charge element within a distribution.

• General Superposition Equation:
  $\mathbf{A}_{total} = \int d\mathbf{A}$

  • Use Case: Summing independent vector contributions (like electric or magnetic fields) from continuous sources.

Fundamental Constants and Symbol Values

• Permittivity of Free Space ($\epsilon_0$):

  • Value: $\approx 8.854 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$

  • Role: Describes the capability of a vacuum to permit electric field lines.

• Unit Vector ($\hat{\mathbf{r}}$):

  • Definition: $\frac{\vec{\mathbf{r}}}{r}$

  • Role: Indicates direction from the charge element $dq$ to the point of observation $P$.

• Distance ($r$): Represents the scalar distance $|\vec{\mathbf{r}}<em>{source} - \vec{\mathbf{r}}</em>{field}|$.

Charge Density Mapping

Setup Equations for Continuous Distributions

To calculate total charge $Q$ or electric field $\mathbf{E}$, use the following substitutions for $dq$:

1. Linear Distributions: $dq = \lambda(l) dl$

2. Surface Distributions: $dq = \sigma(A) dA$

3. Volume Distributions: $dq = \rho(V) dV$

• Electric Field Vector Integral: $\mathbf{E} = \frac{1}{4\pi\epsilon_0} \int \frac{dq}{r^2} \hat{\mathbf{r}}$

  • Strategy: Resolve $\hat{\mathbf{r}}$ into components ($dE<em>x, dE</em>y, dE_z$) based on the symmetry of the object.

Standard Field Results and Use Cases

1. Infinite Line of Charge:

   • Equation: $E = \frac{\lambda}{2\pi\epsilon_0 r}$

   • Use Case: Approximation for the field near long power lines or coaxial cable centers.

2. Infinite Plane of Charge:

   • Equation: $E = \frac{\sigma}{2\epsilon_0}$

   • Use Case: Analyzing the uniform field between the plates of a parallel-plate capacitor.

3. Ring of Charge (On-Axis):

   • Equation: $E<em>z = \frac{1}{4\pi\epsilon</em>0} \frac{Qz}{(z^2 + R^2)^{3/2}}$

   • Symbol Definitions: $z$ is the distance from the ring center; $R$ is the ring radius.

   • Use Case: Particle accelerator beam focusing elements.

Example Solutions for Variable Distributions

• Scenario: A 1D rod along the x-axis from $0$ to $L$ with position-dependent density $\lambda(x) = kx^2$.

• Equation for Total Charge:
  $Q = \int<em>{0}^{L} kx^2 dx = k \left[ \frac{1}{3}x^3 \right]</em>{0}^{L} = \frac{1}{3}kL^3$

• Scenario: A uniform disc of radius $R$ and surface charge $\sigma$.

• Equation for dq:
  Using polar coordinates, the area element is $dA = (2\pi r)dr$, so $dq = \sigma(2\pi r) dr$.