Explain Poisson's Equation
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Answers
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. ... It is a generalization of Laplace's equation, which is also frequently seen in physics.
Step 1: Separate VariablesEdit. Consider the solution to the Poisson equation as u ( x , y ) = X ( x ) Y ( y ) . ...
Step 2: Translate Boundary ConditionsEdit. As in the solution to the Laplace equation, translation of the boundary conditions yields: ...
Step 3: Solve Both SLPsEdit. ...
Step 4: Solve Non-homogeneous EquationEdit.
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Answer:
Poisson's equation
We have seen that the electric field generated by a set of stationary charges can be written as the gradient of a scalar potential, so that
\begin{displaymath}
{\bf E} = - \nabla \phi.
\end{displaymath} (218)
This equation can be combined with the field equation (213) to give a partial differential equation for the scalar potential:
\begin{displaymath}
\nabla^2 \phi = -\frac{\rho}{\epsilon_0}.
\end{displaymath} (219)
This is an example of a very famous type of partial differential equation known as Poisson's equation.
In its most general form, Poisson's equation is written
\begin{displaymath}
\nabla^2 u = v,
\end{displaymath} (220)
where $u({\bf r})$ is some scalar potential which is to be determined, and $ v({\bf r})$ is a known ``source function.'' The most common boundary condition applied to this equation is that the potential $u$ is zero at infinity. The solutions to Poisson's equation are completely superposable. Thus, if $u_1$ is the potential generated by the source function $v_1$, and $u_2$ is the potential generated by the source function
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