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Problem solving - Flux and Gauss' law

         

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

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

The value of electrostatic constant is \(\displaystyle k=\frac{1}{4\pi\varepsilon_0}=8.99 \times 10^9 \text{ N}\cdot\text{m}^2\text{/C}^2.\)

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

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

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

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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