7.5.1 Find the magnetic field between the plates and also above and below them.
7.5.2 Find the magnetic force per unit area on the upper plate, including its direction.
7.5.3 At what speed v would the magnetic force balance the electrical force?
7.6 Show that the magnetic field of an infinite solenoid runs parallel to the axis regardless of the cross-sectional shape of the coil, as long as the shape is constant throughout the length of the solenoid.
7.7 It may have occurred to you that since parallel currents attract, the current within a single wire should contract into a tiny concentrated stream along the axis. Yet in practice the current typically distributes itself quite uniformly over the wire. How do you account for this? If the positive charges (\(\rho_+\)) are at rest, and the negative charges \(\rho_-\)move at speed v, show that \(\rho_- = -\rho_+ + \gamma^2\), where \(\gamma = 1/\sqrt{1-(v/c)^2}\), and \(c^2 = \mu_0 \epsilon_0\).
7.8 A current I flows to the right through a rectangular bar of conducting material, in the presence of a uniform magnetic field B pointing out of the page.
7.8.1 If the moving charges are positive, in which direction are they deflected by the magnetic field? This deflection results in an accumulation of charge on the upper and lower surfaces of the bar, which in turn produce an electric field to counteract the magnetic one. This is known as the Hall Effect.
7.8.2 Find the resulting potential difference (known as the Hall voltage) between the top and bottom of the bar, in terms of B, v, and the dimensions of the bar.
7.8.3 How would your analysis change if the moving charges were negative?