Electric Charge Distribution

• Definition of Charge Distribution: The distribution of electric charge can be classified as uniform & continuous or non-uniform & discontinuous.

  • Uniform and Continuous Charge Distribution: Charge is evenly distributed across the entire object or area.

  • Non-Uniform and Discontinuous Charge Distribution: Charge density varies throughout the object.

Types of Charge Distribution

1. Line Charge Distribution

   • Description: Charge is distributed along a line.

   • Charge Density: Denoted as $\lambda$ (lambda), defined as $\lambda = \frac{dQ}{dL}$ where:

     • $dQ$ = infinitesimal amount of charge

     • $dL$ = infinitesimal length

   • Units: C/m

2. Surface Charge Distribution

   • Description: Charge spread uniformly over a surface.

   • Surface Charge Density: Denoted as $\sigma$, defined as $\sigma = \frac{dQ}{dA}$ where:

     • $dQ$ = infinitesimal amount of charge

     • $dA$ = infinitesimal area

   • Units: C/m²

3. Volume Charge Distribution

   • Description: Charge exists within a volume.

   • Volume Charge Density: Denoted as $\rho$, defined as $\rho = \frac{dQ}{dV}$ where:

     • $dQ$ = infinitesimal amount of charge

     • $dV$ = infinitesimal volume

   • Units: C/m³

Electric Dipole

• Definition of Electric Dipole: An electric dipole consists of two equal but opposite charges separated by a distance.

• Electric Dipole Moment (P):

  • Defined as: $\mathbf{P} = Q \mathbf{d}$ where:

    • $Q$ = magnitude of charge

    • $\mathbf{d}$ = distance vector from negative to positive charge

    • Direction: From negative charge to positive charge.

Electric Potential

• Definition: The electric potential (V) is the work done in bringing a unit positive charge from infinity to a point in space without acceleration.

• Formula: $V = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}$ where:

  • $V$ = electric potential

  • $Q$ = charge creating the potential

  • $r$ = distance from the charge

  • $\varepsilon_0$ = permittivity of free space

Electric Field for Infinite Line Charge

• Electric Field (E): For a uniformly charged infinite line, the electric field is calculated using: $E = \frac{1}{2\pi \varepsilon_0 \cdot r}$ where:

  • $r$ = perpendicular distance from the line charge to the point of field measurement.

Problem Example

• Given a charge density $\rho = 10$ C/m distributed along an infinite line, calculate the electric field at point P(√2, √23, 2):

  • $E = \frac{\rho}{2\pi \varepsilon_0 r}$

Surface Charge Density and Electric Field Calculation

• Example Calculation:

  • For a point at distance $r = 20$ cm:

  • $E = \frac{P<em>s}{\varepsilon</em>0}$

  • Given $P_s = 10$ C/m², calculate field intensity at $x = 4$ in free space.

• Using Surface Charge Density (PS):
  $P_s = \frac{Q}{A}$

• Continuing Calculations:

  • Compute electric field due to various configurations.

Calculation of Electric Field from Point Charges

• Example: Electric field intensity at point (3, 0, 10) due to a point charge located at different coordinates.

  • Field calculated as $E = \frac{k \cdot |Q|}{r^2}$ where:

  • $k$ = Coulomb's constant

  • $Q$ = point charge

  • $r$ = distance from charge to field point.

Biot-Savart Law

• Definition: The Biot-Savart Law describes the magnetic field generated by an electric current.

  • Mathematical Expression:
    $dB = \frac{\mu_0}{4\pi} \frac{I \, dL \times \mathbf{r}}{| extbf{r}|^3}$ where:

  • $dB$ = elemental magnetic field

  • $I$ = current

  • $dL$ = current element

  • $\mathbf{r}$ = position vector

  • $\mu_0$ = permeability of free space

• Implications of Biot-Savart Law:

  • Allows calculation of magnetic fields produced by current-carrying conductors.

  • Useful for determining fields in complex geometries.

Ampere's Circuital Law

• Statement: The line integral of the magnetic field intensity (H) around a closed loop is equal to the total current (I) passing through that loop:

  $\oint \mathbf{H} \cdot d\mathbf{l} = I_{enc}$

• Applications:

  • Useful for calculating magnetic fields in situations with a high degree of symmetry.

Summary of Formulas:

Charge Density:

• Line: $\lambda = \frac{dQ}{dL}$

• Surface: $\sigma = \frac{dQ}{dA}$

• Volume: $\rho = \frac{dQ}{dV}$

Electric Potential:

• $V = K \frac{Q}{r}$

Electric Field from Line Charge:

• $E = \frac{\lambda}{2\pi \varepsilon_0 r}$

Biot-Savart Law:

• $dB = \frac{\mu_0}{4\pi} \frac{I \, dL \times \mathbf{r}}{| extbf{r}|^3}$

Ampere's Circuital Law:

• $\oint \mathbf{H} \cdot d\mathbf{l} = I_{enc}$