What does electric flux measure for a surface in an electric field?

What is the direction of the area vector for a flat surface?

Write the flux through a flat surface in a uniform electric field in terms of \(E\), \(A\), and \(\theta\).

A \(60\,\mathrm{N\,C^{-1}}\) uniform electric field is perpendicular to a \(0.40\,\mathrm{m^2}\) surface and points with the chosen area vector. Find \(\Phi_E\).

A uniform electric field is parallel to a flat surface. What is the electric flux through the surface?

A \(120\,\mathrm{N\,C^{-1}}\) field makes a \(60^\circ\) angle with the area vector of a \(0.50\,\mathrm{m^2}\) surface. Find the flux.

A \(90\,\mathrm{N\,C^{-1}}\) field points opposite the area vector of a \(0.20\,\mathrm{m^2}\) flat surface. Find the flux.

A flat surface has area vector \(\vec A=(0.30\,\mathrm{m^2})\hat\imath\). A uniform field is \(\vec E=(50\,\mathrm{N\,C^{-1}})\hat\imath\). Find \(\Phi_E\).

A flat surface has \(\vec A=(0.20\hat\imath+0.10\hat\jmath)\,\mathrm{m^2}\) and \(\vec E=(40\hat\imath-30\hat\jmath)\,\mathrm{N\,C^{-1}}\). Find the flux.

A circular plate of radius \(0.15\,\mathrm{m}\) has its area vector \(35^\circ\) from a \(200\,\mathrm{N\,C^{-1}}\) uniform field. Find the flux.

A square loop of side \(0.30\,\mathrm{m}\) is rotated from having its area vector parallel to a uniform field to having its area vector perpendicular to the field. If \(E=75\,\mathrm{N\,C^{-1}}\), what is the change in flux?

A \(0.10\,\mathrm{m^2}\) surface has flux \(4.0\,\mathrm{N\,m^2\,C^{-1}}\) in a \(100\,\mathrm{N\,C^{-1}}\) uniform field. What angle does the field make with the area vector?

A uniform field \(\vec E=E\hat\imath\) passes through a cube of side \(L\). Find the flux through the right face, left face, and the total flux through the closed cube.

A uniform electric field points upward through a horizontal disk. Compare the flux if the disk's area vector is chosen upward versus downward.

A flat rectangular surface has side lengths \(a\) and \(b\). Its normal is tilted by angle \(\theta\) from a uniform field \(\vec E\). Derive the flux and state when its magnitude is largest.

A curved surface is divided into small patches. Explain why the flux is \(\int_S \vec E\cdot d\vec A\), not simply \(EA\).

A closed box is placed in a uniform electric field. Without using Gauss's law, argue from area vectors why the total flux through the box is zero.

A flat surface has fixed area \(A\) in a uniform field \(E\). You may choose its orientation. Derive the range of possible flux values and explain the geometric meaning of the endpoints.

Two flat surfaces have the same boundary curve but one is bulged outward. In a uniform electric field, explain why their fluxes need not be the same if the boundary is not closed by an additional surface.

A surface is made from many small flat patches with area vectors \(\Delta\vec A_i\). Starting from the uniform-field dot product, derive the surface-integral definition of electric flux and state the assumptions needed for the limiting process.