Cube The cube in Fig. 24-25 has edge lengths of 1.40 m and is oriented as shown with its bottom

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Cube The cube in Fig. 24-25 has edge lengths of 1.40 m and...

Cube The cube in Fig. $24-25$ has edge lengths of $1.40 \mathrm{~m}$ and is oriented as shown with its bottom face in the $x-y$ plane at $z=$ $0.00 \mathrm{~m}$. Find the electric flux through the right face if the uniform electric field, in newtons per coulomb, is given by (a) $6.00 \hat{i}$, (b) $-2.00 \hat{j}$, and (c) $-3.00 \hat{\mathrm{i}}+$ $4.00 \hat{k}$. (d) What is the total flux through the cube for each of these fields?

Cube The cube in Fig. $24-25$ has edge lengths of $1.40 \mathrm{~m}$ and is oriented as shown with its bottom face in the $x-y$ plane at $z=$ $0.00 \mathrm{~m}$. Find the electric flux through the right face if the uniform electric field, in newtons per coulomb, is given by (a) $6.00 \hat{i}$,
(b) $-2.00 \hat{j}$, and
(c) $-3.00 \hat{\mathrm{i}}+$
$4.00 \hat{k}$. (d) What is the total flux through the cube for each of these fields?

Key Concepts

Electric Flux

Electric flux is a measure of the number of electric field lines passing through a given surface. It is mathematically defined as the surface integral of the electric field over that surface, commonly expressed as ? = ? E · dA, where E is the electric field vector and dA is the differential element of area with an outward normal vector. This concept is central to understanding how charges influence the space around them and is used in various physical laws.

Uniform Electric Field

A uniform electric field is one in which the magnitude and direction of the field are the same at every point in space. This simplification allows for straightforward calculations because the electric field vector can be taken out of the integral when computing the flux, and the flux through a flat surface can then be calculated by simply multiplying the field component normal to the surface by the area.

Surface Integral

The surface integral is a mathematical tool used for integrating a function over a surface in space. When applied to electric fields, it involves summing the contributions of the field over each differential area element. This approach is essential in computing electric flux, as it takes into account the orientation of each area element relative to the electric field direction.

Gauss's Law

Gauss's Law states that the total electric flux through a closed surface is proportional to the net charge enclosed within that surface. Mathematically, it is given by ?_total = Q_enclosed/??, where Q_enclosed is the net charge inside the closed surface and ?? is the permittivity of free space. This law is a powerful tool in electrostatics, particularly in situations with high symmetry, though it also implies that if no charge is enclosed, the net flux through a closed surface is zero.

Vector Dot Product

The vector dot product is a mathematical operation that takes two vectors and returns a scalar. In the context of electric flux, it is used to calculate the component of the electric field that is normal to a surface. The dot product between the electric field vector and the area vector (which is oriented perpendicular to the surface) determines how much of the electric field is effectively 'passing through' the surface, which is essential for flux calculations.

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