Electric flux and Gauss’s law

A disk of radius $10\mathrm{cm}$ is in a uniform electric field of magnitude $6.0\cdot 10^{3}N/C$, and the normal to the disk makes an angle of $30^{\circ }$ with the field. Compute the flux of the field through the disk.

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A hemispherical surface of radius $R=0.20\mathrm{m}$, without its base, is immersed in a uniform electric field of magnitude $5.0\cdot 10^{3}N/C$, parallel to the axis of symmetry of the hemisphere and directed from the plane of the base toward the top. Compute the flux of the field through the hemispherical surface, oriented with the normal pointing toward the convex side.

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The outward electric flux through a closed surface is $-4.5\cdot 10^3\mathrm{N}\cdot {\mathrm{m}}^2/C$. Compute the total charge contained in the surface.

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A point charge of $3.0\mathrm{nC}$ is at the center of a cube of edge $20\mathrm{cm}$. Compute the outward electric flux through one face of the cube.

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An insulating sphere of radius $4.0\mathrm{cm}$ has the charge $12\mathrm{nC}$ distributed uniformly throughout its volume. Compute the magnitude of the electric field at $1.0\mathrm{cm}$ and at $6.0\mathrm{cm}$ from the center.

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A thin spherical shell of radius $10\mathrm{cm}$ carries the charge $5.0\mathrm{nC}$ distributed uniformly. Compute the magnitude of the electric field at $5.0\mathrm{cm}$ and at $20\mathrm{cm}$ from the center.

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At $20\mathrm{cm}$ from a very long straight wire, charged uniformly, the electric field has magnitude $900N/C$ and points away from the wire. Compute the linear charge density of the wire.

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A very long insulating cylinder of radius $3.0\mathrm{cm}$ is charged uniformly with density $2.0\mu C/{\mathrm{m}}^3$. Compute the magnitude of the electric field at $1.5\mathrm{cm}$ and at $9.0\mathrm{cm}$ from the axis.

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A very large plane is charged uniformly with density $5.0nC/{\mathrm{m}}^2$. Compute the magnitude of the electric field near the plane and the acceleration of an electron in the field, neglecting its weight.

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Two very large parallel planes have uniform surface densities $+\sigma$ and $-\sigma$. Between the planes, far from the edges, the electric field has magnitude $2.0\cdot 10^{4}N/C$. Compute $\sigma$ and the field outside the planes.

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A very large insulating slab of thickness $4.0\mathrm{cm}$ is charged uniformly with density $1.0\mu C/{\mathrm{m}}^3$. Compute the magnitude of the electric field outside the slab and at $1.0\mathrm{cm}$ from the midplane.

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In a region of space the electric field is $\vec{E}=a(x\hat{ı}+y\hat{ȷ}+z\hat{k})$, with $a=1.5\cdot 10^{4}N/(\mathrm{C}\cdot \mathrm{m})$. Compute the charge density in the region.

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An insulating sphere of radius $R=5.0\mathrm{cm}$ is charged with density $\rho (r)={\rho }_0\frac{r}{R}$, with ${\rho }_0=4.0\mu C/{\mathrm{m}}^3$. Compute the total charge, the field inside as a function of $r$ and its magnitude at the surface.

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