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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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