(a) In Equation 9.21, we asserted a relation between the...

(a) In Equation $9.21,$ we asserted a relation between the divergence of the electric field and the charge density. Imitate the one-dimensional derivation culminating in Equation 9.20 to deduce Equation $9.21 .$ To do this, explicitly compute the flux through a small cubical volume element like that shown in Figure 9.9 (b) By using the fact that the electric field can be written as $\mathbf{E}(\mathbf{r})=-\nabla V(\mathbf{r}),$ write a partial differential equation relating the potential and the charge density.

(a) In Equation $9.21,$ we asserted a relation between the divergence of the electric field and the charge density. Imitate the one-dimensional derivation culminating in Equation 9.20 to deduce Equation $9.21 .$ To do this, explicitly compute the flux through a small cubical volume element like that shown in Figure 9.9
(b) By using the fact that the electric field can be written as $\mathbf{E}(\mathbf{r})=-\nabla V(\mathbf{r}),$ write a partial differential equation relating the potential and the charge density.

Key Concepts

Gauss’s Law in Differential Form

This concept expresses the relationship between the electric field and the charge density at a point in space through the equation ? · E = ?/??. It is derived by applying the divergence theorem to a small volume so that the net flux out of the volume is directly linked to the total charge enclosed, bridging the integral formulation of Gauss’s law with a local (pointwise) differential statement.

Divergence

Divergence measures the magnitude of a source or sink at a given point in a vector field. By computing the net flux through the faces of an infinitesimal cubical volume element, one can quantify how much the electric field diverges from that point, thereby relating the field's behavior to the local charge density.

Electric Potential

The electric potential is a scalar field defined such that its negative gradient yields the electric field (E = -?V). This relation simplifies many electrostatic problems by converting a vector field problem into one involving a scalar function, which is often easier to handle mathematically.

Poisson’s Equation

By substituting E = -?V into Gauss’s law, one derives Poisson’s equation, ?²V = -?/??. This partial differential equation directly relates the Laplacian of the electric potential to the charge density, providing a fundamental tool for solving electrostatic problems where the charge distribution is known.

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