## By Pavel Panchekha, Jeffrey Prouty

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

## Level I

1. 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.

1. 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?
2. Find and explain any forces on the projectile. Ignore gravity.
3. 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

1. 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?
2. 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

1. 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.