In one dimension, what is the relation between \(E_x\) and \(V(x)\)?

Does the electric field point toward increasing or decreasing electric potential?

If \(V\) is constant throughout a region, what is \(\vec E\) there?

Potential decreases by \(30\,\mathrm{V}\) over \(0.50\,\mathrm{m}\) in the \(+x\) direction. Estimate \(E_x\).

A uniform electric field \(200\,\mathrm{V\,m^{-1}}\) points in \(+x\). Find \(\Delta V\) for a \(0.10\,\mathrm{m}\) displacement in \(+x\).

For \(V(x)=12-4x\) with \(x\) in meters and \(V\) in volts, find \(E_x\).

For \(V(x)=ax^2\), derive \(E_x(x)\) and state where the electric field is zero.

For \(V(x,y)=Axy\), find \(\vec E\).

A one-dimensional potential is \(V(x)=V_0e^{-x/\ell}\). Derive \(E_x(x)\), then determine where the field magnitude is largest for \(x\ge0\).

A potential has the form \(V(x,y)=A(x^2-y^2)\). Derive \(\vec E\), find the points where \(\vec E=0\), and describe the directions of the field on the \(x\)-axis and \(y\)-axis.