Set up the integal analogous to that used in the proof of Eq. 30.-6, with different limits of integration.
Use the result of 30-17P with the substitution of a for L and a/2 for R. Use superposition of four sides.
Consider an infinitesimal section of the ribbon of thickness dx located a distance x from the point of evaluation P; the contribution to dB is upwards for each infinitesimal section. Integrate from d to d+w to pick up each contribution dB.
This is a good geometry for checking that the answer reduces to a current line when d becomes large compared to w!
The forces on the two sides of length b balance. Evaluate the other two components and sum.
a) Use the result for the magnetic field about a long straight wire; use superposition to find the magnetic field at each point on the projectile. Integrate along the length of the projectile to get the total force on it.
b) The work done by the magnetic force will go into kinetic energy of the projectile.