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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.

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 \(\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.\)

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

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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 \(\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.\)

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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 \(\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.\)

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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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