Poisson's equation is a cornerstone of electrostatics, establishing a direct link between the electric potential and the distribution of electric charges in space. This equation can be meticulously derived from Gauss's Law in its differential (point) form, unveiling a fundamental relationship in the realm of electromagnetism.
Step-by-Step Breakdown of the Derivation
1. Gauss's Law in Integral Form
Gauss's Law, in its most general form, describes the relationship between the electric flux through a closed surface and the net charge enclosed within that surface.
The integral form of Gauss's law is given by:
\[ \oint_{S} \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \]
Where:
- \(\oint_{S} \vec{E} \cdot d\vec{A}\): Electric flux through the closed surface \(S\).
- \(\vec{E}\): Electric field vector.
- \(d\vec{A}\): Differential area vector, pointing outward.
- \(Q_{\text{enc}}\): Total charge enclosed within the volume \(V\).
- \(\varepsilon_0\): Permittivity of free space ≈ \(8.85 \times 10^{-12}\ \text{F/m}\).
2. Expressing Enclosed Charge in Terms of Charge Density
For a continuous distribution of charge within a volume, the total enclosed charge \(Q_{enc}\) can be expressed as the volume integral of the charge density \(\rho \) over the enclosed volume
\[ Q_{\text{enc}} = \int_V \rho\, dV \]
Where:
- \(\rho\): Volume charge density, representing the charge per unit volume at a given point in space
- \(dV\): Infinitesimal volume element.
3. Applying the Divergence Theorem
The Divergence Theorem connects a surface integral to a volume integral, stating the flux of a vector field through a closed surface equals the volume integral of the divergence of the field over the enclosed volume.
\[ \oint_S \vec{E} \cdot d\vec{A} = \int_V (\nabla \cdot \vec{E})\, dV \]
Here, \(\nabla \cdot \vec{E}\) is the divergence of the electric field.
4. Deriving Gauss's Law in Point Form
Combining previous results, by equating the surface integral of the electric field (flux) with the volume integral of the charge density and applying the Divergence Theorem, the result is:
\[ \int_V (\nabla \cdot \vec{E})\, dV = \frac{1}{\varepsilon_0} \int_V \rho\, dV \]
Since this holds for any volume \(V\):
\[ \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0} \]
This is Gauss's Law in differential (point) form.
5. Relating Electric Field to Electric Potential
In electrostatics, the electric field (\(\vec{E}\)) is the negative gradient of the electric potential (\(V\))
\[ \vec{E} = -\nabla V \]
This means the electric field points in the direction of decreasing potential.
6. Substituting into Gauss's Law
Substitute into the point form of Gauss's Law:
\[ \nabla \cdot (-\nabla V) = \frac{\rho}{\varepsilon_0} \]
7. Introducing the Laplacian Operator
The divergence of a gradient is the Laplacian:
\[ \nabla^2 V = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} \]
So:
\[ -\nabla^2 V = \frac{\rho}{\varepsilon_0} \]
8. Final Form of Poisson's Equation
Rewriting gives:
\[ \nabla^2 V = -\frac{\rho}{\varepsilon_0} \]
Special Case: Laplace's Equation
Poisson's equation becomes Laplace's Equation when \(\rho = 0\):
\[ \nabla^2 V = 0 \]
This is Laplace’s Equation, applicable in charge-free regions.
Conclusion
This derivation elegantly connects Gauss's Law and electric potential to form Poisson's Equation — a vital analytical tool in solving electrostatic field problems.
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