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Poisson equation
Poisson Equation: $$\nabla \cdot \overrightarrow{E} = \frac{\rho}{\varepsilon_0}$$ where: ∇ is the vector operator representing the divergence of a ve...
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Poisson Equation: $$\nabla \cdot \overrightarrow{E} = \frac{\rho}{\varepsilon_0}$$ where: ∇ is the vector operator representing the divergence of a ve...
Poisson Equation:
where:
∇ is the vector operator representing the divergence of a vector field.
E is the electric field vector.
ρ is the charge density.
ε₀ is the vacuum permittivity.
Interpretation:
This equation states that the net divergence of the electric field must equal the charge density divided by the vacuum permittivity. In other words, the electric field must satisfy a balance between the sources (charges) and the empty space filling the region.
Applications:
The Poisson equation has diverse applications in electrostatics, including:
Electric field distribution: It helps determine the electric field distribution in a charge-free region or a uniformly charged sphere.
Electrostatic potential: The equation connects the electric potential and the charge density, allowing us to calculate the potential due to a point charge or a distribution of charges.
Dielectric analysis: It is used in the analysis of dielectric materials, where the electric field distribution depends on the permittivity of the medium.
Magnetic fields: Poisson's equation can be generalized to relate the magnetic field vector to the current density.
Examples:
In a uniform charge distribution, the electric field lines are equally spaced and point towards the charges.
In a point charge, the electric field is infinitely directed towards the point charge, with the magnitude proportional to the charge magnitude.
In a dielectric, the electric field lines are bent around the boundary of the material, indicating the electric field intensity decreases with distance from the boundary