(II) In a certain region of space, the electric field is...

(II) In a certain region of space, the electric field is constant in direction (say horizontal, in the $x$ direction), but its magnitude decreases from $E=560 \mathrm{~N} / \mathrm{C}$ at $x=0$ to $E=410 \mathrm{~N} / \mathrm{C}$ at $x=25 \mathrm{~m} .$ Determine the charge within a cubical box of side $\ell=25 \mathrm{~m},$ where the box is oriented so that four of its sides are parallel to the field lines (Fig. $22-28$ ). $x=0$ $x=25 \mathrm{~m}$

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the net electric flux through a closed surface to the total charge enclosed by that surface. It is often mathematically expressed as ?E · dA = Q_enc/??, meaning that the integral of the electric field over a closed surface is proportional to the enclosed charge. This law is instrumental in solving problems where symmetry allows one to easily compute the flux and thereby the charge distribution.

Electric Flux

Electric flux quantifies the total number of electric field lines passing through a given surface. In the context of Gauss's Law, the flux through a closed surface is directly linked to the charge enclosed by that surface. Understanding flux is key in determining how variations in the electric field across a surface contribute to a net charge within a region.

Divergence Theorem

The divergence theorem bridges the relationship between the flux of a vector field through a closed surface and the divergence of the field within the volume bounded by that surface. This theorem underpins Gauss's Law by converting a surface integral into a volume integral, allowing for analysis of how local changes in the electric field (its divergence) indicate the presence of charge density in a region.

Non-uniform Electric Field

A non-uniform electric field is one whose magnitude (and potentially direction) varies in space. In many physical situations, such variations require careful application of Gauss's Law, as the standard symmetric assumptions used to simplify the flux calculations may no longer hold. The analysis of non-uniform fields involves accurately accounting for the variations across the surfaces of interest to determine the corresponding effects on charge distribution.

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