The Green function method for solution of the Poisson equation with different types of boundary conditions, Dirichlet and Neuman, are discussed. $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\Prime}{^\prime}$
$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}$ Problems with cylindrical symmetry can be solved by separating the variables of the Poisson equation in cylindrical coordinates. The separation of variables for this class of problems and boundary conditions are explained.
The electric potential due to charge spread uniformly on a finite line segment is computed.The electric potential due to charge spread uniformly on a finite line segment is computed.
The equation of continuity appears in different branches of physics. It represents a local conservation law. In order to be consistent with requirement if special relativity every conserved quantity must come with a current which gives the flow of the conserved quantity across a surface and the two must obey equation of continuity. Taking the example of momentum conservation, we briefly discuss the interpretation of stress tensor giving flow of momentum per unit time as a surface integral. The surface integral, in turn, gives the force on the surface.
The electrostatic energy of a continuous charge distribution is defined as the energy required to assemble the charges at infinity into the positions as in the given distribution. For a continuous charge distribution it is shown to be \( \dfrac{\epsilon_0}{2}\iiint(\vec E\cdot\vec E) dV\) . Thus a volume of space having nonvanishing electric field has energy density \(\dfrac{\epsilon_0}{2}(\vec E\cdot\vec E)\).The expression for the electrostatic energy reduces to the usual answer \(\frac{1}{2} CV^2\) for a charged parallel plate capacitor. For a uniformly charged sphere of radius \(R\) the electrostatic energy is proved to be equal to \(\frac{3}{5}\Big(\frac{Q^2}{4\pi\epsilon_0 R^2} \Big)\).
The concept of electric potential for static electric field is defined as work done on a unit charge. The expression for the electric potential of a \(q\) charge is obtained. For a system of point charges the potential can be written down as superposition of potential due to individual charges. As an illustration we compute potential due to a dipole.
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