Gauss' law

         

A Gaussian cube of edge length 1.0 m1.0\text{ m} is located in the xyzxyz-space, as shown in the above figure. If the electric field in this space is expressed as E=5.0xi^+4.0j^ N/C,\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 ε0=8.85×1012 C2/Nm2.\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 m27\text{ m} from the bottom of the device the magnitude of electric field is 50.0 N/C,50.0 \text{ N/C}, and at a height of 18 m18\text{ m} from the bottom the magnitude of electric field is 80.0 N/C.80.0\text{ N/C}. If the bottom face of a cube with side length 9 m9\text{ m} is at a height of 18 m18\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 ε0=8.85×1012 C2/Nm2.\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.S. The cross section of a 33-D Gaussian surface SS is indicated as green ellipse. If q1=q4=+3.9 nC,q_1=q_4=+3.9\text{ nC}, q2=q5=5.3 nC,q_2=q_5=-5.3\text{ nC}, and q3=q6=3.9 nC,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 ε0=8.85×1012 C2/Nm2.\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.

Consider an isosceles right triangle ABCABC of base d,d, as shown in the above figure. A proton is at a distance of d2\frac{d}{2} directly above the midpoint of the hypotenuse AB.\overline{AB}. What is the magnitude of the electric flux through the triangle?

The charge of a proton is q=+1.6×1019 Cq=+1.6 \times 10^{-19}\text{ C} and the value of the permittivity constant is ε0=8.85×1012 C2/Nm2.\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 yy-axis, as shown in the above figure. At each point on the surface of the cylinder, the electric field is parallel to the yy-axis. The area of the base of the cylinder is 9.0 m29.0 \text{ m}^2 and the height of the cylinder is 5.0 m.5.0 \text{ m}. On the top face of the cylinder the field is E=42.0j^ N/C,\overrightarrow{E}=-42.0 \hat{j}\text{ N/C}, and on the bottom face it is E=+22.0j^ N/C.\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 ε0=8.85×1012 C2/Nm2.\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.

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