# Problem Set 12 (Given 02/12/12)

## Level I

- A constant \(B\) field is defined by \(\mathbf{B} = B_0 {\hat z}\). An electron moving with velocity \(\mathbf{v} = v_x {\hat x} + v_y {\hat y} + v_z {\hat z}\). As discussed in class, the electron will move in a helix (spiral). Find how far the electron will move in one rotation around its spiral.

A railgun works as follows. Two long, parallel rails are made of a conductive metal, and a conductive projectile is placed on them so that it can move along the rails while maintaining electric contact. The other ends of the rails are connected to a voltage source. As current flows along the rails, it creates a magnetic field between the rails.

- Find the magnetic field in the middle of the two rails if the
current along the rails is \(I\) and the distance between the rails
is \(2r\). What is the magnitude
*and direction*of the \(B\) field? - Find and explain any forces on the projectile. Ignore gravity.
- The resistance of steel, as might be used for the rails, is \(2 \cdot 10^-5 \Omega / m\) (that is, per meter of rail). If the rails are ten meters long, and the current one mega-ampere, find the energy lost to heat.

## Level II

- Two long parallel wires have current \(I\) flowing in the same direction along them. Each wire lies along the center of a pipe of radius \(r\). The two wires are \(d\) distance apart. What is the average magnetic field between the two wires in their plane?
- A wire consists of two parallel half-infinite wires connected by a half-circle. Using the Biot-Savart law, find the magnetic field in the center of the half-circle.

## Level III

- The Biot-Savart law allows us to find magnetic field contribution of an infinitesimal piece of wire. On the other hand, recall that in class we found the magnetic field due to a plane of current; we couldn't use the plain Biot-Savart law for this job, since it is a piece of plane, not wire. By integrating in the correct way, derive a modified form of the Biot-Savart law suitable for finding the magnetic field due to a surface along which current flows.