What component of \(\vec E\) contributes to electric flux through a surface?

Write the flux element \(d\Phi_E\) in terms of \(\vec E\) and \(d\vec A\).

A surface patch has \(d\vec A=(0.020\,\mathrm{m^2})\hat\jmath\) and \(\vec E=(150\,\mathrm{N\,C^{-1}})\hat\jmath\). Find \(d\Phi_E\).

A surface patch has \(d\vec A=(0.030\,\mathrm{m^2})\hat k\) and \(\vec E=(80\,\mathrm{N\,C^{-1}})\hat\imath\). Find \(d\Phi_E\).

A rectangular panel has area \(0.60\,\mathrm{m^2}\) and outward normal \(\hat\imath\). The uniform field is \(\vec E=(25\hat\imath+10\hat\jmath)\,\mathrm{N\,C^{-1}}\). Find the flux.

A rectangular panel has area \(0.40\,\mathrm{m^2}\) and outward normal \(-\hat\jmath\). The uniform field is \(\vec E=(12\hat\imath+30\hat\jmath)\,\mathrm{N\,C^{-1}}\). Find the flux.

A cube of side \(0.20\,\mathrm{m}\) is in a uniform field \(\vec E=(100\,\mathrm{N\,C^{-1}})\hat\imath\). Find the flux through the face whose outward normal is \(+\hat\imath\).

For the same cube and field as question 7, find the total flux through the closed cube.

A cylinder's axis is parallel to a uniform field \(\vec E\). Which parts of the cylinder have nonzero flux: the two circular end caps, the curved side, or both?

A cylinder of radius \(0.10\,\mathrm{m}\) has its axis along a \(300\,\mathrm{N\,C^{-1}}\) uniform field. Find the flux through the end cap whose outward normal points with the field.

The same cylinder has two end caps and a curved side. Find the total flux through the closed cylinder in the uniform axial field.

A rectangular box has side lengths \(a\), \(b\), and \(c\). It is placed in uniform field \(\vec E=E\hat k\). Find the flux through each pair of opposite faces and the total flux.

A square plate of side \(0.25\,\mathrm{m}\) has unit normal \(\hat n=(0.60\hat\imath+0.80\hat\jmath)\). The field is \(\vec E=(50\hat\imath)\,\mathrm{N\,C^{-1}}\). Find the flux.

A closed rectangular box extends from \(x=0\) to \(x=L\), with face area \(A\) perpendicular to the \(x\)-axis. The field is \(\vec E=\alpha x\,\hat\imath\). Compute the total flux through the box.

A closed rectangular box extends from \(x=x_1\) to \(x=x_2\), with cross-sectional area \(A\). The field is \(\vec E=E_0\hat\imath+\beta x\hat\imath\). Compute the total flux through the box.

A sphere is placed in a uniform electric field. Without evaluating an integral in spherical coordinates, explain why the net flux through the sphere is zero.

A closed surface is made of flat panels in a uniform field. Prove from the panel area vectors that the net flux is zero.

A nonuniform field is \(\vec E=\alpha x\hat\imath+\beta y\hat\jmath\). A rectangular box spans \(0<x<a\), \(0<y<b\), \(0<z<c\). Compute the net flux through the box by summing faces.

A field is \(\vec E=E_0\hat z\). A closed surface is cut into a horizontal disk and a curved cap above it. Use flux cancellation to find the flux through the cap if the disk's outward normal is \(-\hat z\).

For a closed polyhedral surface in a slowly varying field, explain how the face-sum method becomes a surface integral and what information is lost if only the net flux is reported.