In which direction does the electric dipole moment \(\vec p\) point?

State the magnitude of the dipole moment for charges \(\pm q\) separated by distance \(d\).

A dipole is aligned with a uniform electric field. Is the torque magnitude zero or nonzero?

Find \(p\) for charges \(\pm 6.0\,\mathrm{nC}\) separated by \(2.0\,\mathrm{mm}\).

A dipole with \(p=3.0\times10^{-8}\,\mathrm{C\,m}\) is in a \(500\,\mathrm{N\,C^{-1}}\) field at \(30^\circ\). Find the torque magnitude.

For a dipole in a uniform field, compare the potential energy at \(\theta=0^\circ\) and \(\theta=180^\circ\). Which is lower?

A dipole is placed in a uniform electric field. Explain why the net force is zero but the net torque can be nonzero.

Far from a dipole, the axial field magnitude is \(2kp/r^3\). If distance doubles, by what factor does this far-field magnitude change?

A dipole moment \(\vec p=p\hat\imath\) is placed in a uniform field \(\vec E=E(\cos\phi\,\hat\imath+\sin\phi\,\hat\jmath)\). Derive \(\vec\tau\) and \(U\), and identify stable and unstable orientations.

A dipole consists of charges \(\pm q\) at \(x=\pm d/2\). For a point on the \(+x\)-axis with \(x\gg d\), use a first-order expansion to show why the axial field scales as \(1/x^3\) rather than \(1/x^2\).