Problem solving - Flux and Gauss' law

         

Consider an infinitely long, very thin metal tube with radius R=2.90 cm.R=2.90\text{ cm}. The above figure shows a section of it. If the linear charge density of the cylinder is λ=1.50×108 C/m,\lambda=1.50 \times 10^{-8} \text{ C/m}, what is the approximate magnitude of the electric field at radial distance r=2R?r=2R?

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 solid sphere of radius a=2.30 cm,a=2.30\text{ cm}, which has a net uniform charge of q=+4.50 fC.q=+4.50\text{ fC}. What is the approximate magnitude of the electric field at radial distances (a) r=a/2r=a/2 and (b) r=a?r=a ?

The value of electrostatic constant is k=14πε0=8.99×109 Nm2/C2.\displaystyle k=\frac{1}{4\pi\varepsilon_0}=8.99 \times 10^9 \text{ N}\cdot\text{m}^2\text{/C}^2.

Consider a metal rod with radius of R1=1.20 mmR_1=1.20\text{ mm} and length L=12.00 m,L=12.00\text{ m}, which is inside a very thin coaxial metal cylinder with radius of R2=10.0R1R_2=10.0 R_1 and length L,L, as shown in the above figure. If the net charge on the rod and on the cylinder is Q1=+3.40×1012 CQ_1=+3.40 \times 10^{-12}\text{ C} and Q2=2.0Q1,Q_2=-2.0 Q_1, respectively, what is the approximate magnitude of the electric field at radial distance r=2.0×R2,r=2.0 \times R_2, assuming that the charge density of both the rod and the cylinder are uniform?

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 Gaussian surface in the shape of a cube with edge length 2.40 m,2.40\text{ m}, as shown in the above figure. If the electric field in which the Gaussian surface lies can be expressed as E=(2.00x+3.00)i^+5.00j^+6.00k^ (N/C),\vec{E}=(2.00x+3.00)\hat{i}+5.00\hat{j}+6.00\hat{k} \text{ (N/C)}, where xx is in meters, what is the approximate net charge contained by 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 infinitely large non-conducting plate of thickness d=9.20 mm.d=9.20\text{ mm}. The above figure shows a cross section of the plate, where the origin of the xx-axis is at the plate's center. If the plate has a uniform charge density of ρ=5.80 fC/m3,\rho=5.80\text{ fC/m}^3, what is the magnitude of the electric field at the xx-coordinate of (a) 4.60 mm4.60\text{ mm} and (b) 27.0 mm?27.0 \text{ mm}?

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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