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## Electric Flux

Electric flux connects the geometry of conductors to the fields they generate. Learn this powerful tool and shortcut your way to the electric field of symmetrical arrangements like wires and sheets.

# Gauss' Law

A Gaussian cube of edge length $$1.0\text{ m}$$ is located in the $$xyz$$-space, as shown in the above figure. If the electric field in this space is expressed as $\overrightarrow{E}=5.0x\hat{i}+4.0\hat{j} \text{ N/C},$ what is the net charge enclosed by the Gaussian cube?

The value of the permittivity constant is $$\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.$$

The electric field in a certain device is vertically downward. At a height of $$27\text{ m}$$ from the bottom of the device the magnitude of electric field is $$50.0 \text{ N/C},$$ and at a height of $$18\text{ m}$$ from the bottom the magnitude of electric field is $$80.0\text{ N/C}.$$ If the bottom face of a cube with side length $$9\text{ m}$$ is at a height of $$18\text{ m}$$ from the bottom of the device, what is the approximate net amount of charge contained in the cube?

The value of the permittivity constant is $$\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.$$

The above figure shows six charged particles, where three of them are enclosed by a Gaussian surface $$S.$$ The cross section of a $$3$$-D Gaussian surface $$S$$ is indicated as green ellipse. If $$q_1=q_4=+3.9\text{ nC},$$ $$q_2=q_5=-5.3\text{ nC},$$ and $$q_3=q_6=-3.9\text{ nC},$$ what is the net electric flux through the Gaussian surface?

The value of the permittivity constant is $$\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.$$

Consider an isosceles right triangle $$ABC$$ of base $$d,$$ as shown in the above figure. A proton is at a distance of $$\frac{d}{2}$$ directly above the midpoint of the hypotenuse $$\overline{AB}.$$ What is the magnitude of the electric flux through the triangle?

The charge of a proton is $$q=+1.6 \times 10^{-19}\text{ C}$$ and the value of the permittivity constant is $$\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.$$

Consider a cylinder base which is perpendicular to the $$y$$-axis, as shown in the above figure. At each point on the surface of the cylinder, the electric field is parallel to the $$y$$-axis. The area of the base of the cylinder is $$9.0 \text{ m}^2$$ and the height of the cylinder is $$5.0 \text{ m}.$$ On the top face of the cylinder the field is $$\overrightarrow{E}=-42.0 \hat{j}\text{ N/C},$$ and on the bottom face it is $$\overrightarrow{E}=+22.0 \hat{j}\text{ N/C}.$$ What is the approximate net charge contained within the cylinder?

The value of the permittivity constant is $$\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.$$

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