DUE October 31

Fermilab Booster Gradient Magnet

  1. Show that for a pure “H”-style dipole magnet with \(N\) turns per pole, current \(I\) per turn, and total gap height \(d\), the central field is given by

\[ B_y \approx \frac{2\mu_0NI}{d} \]

  1. If a magnet has a gap of 50.8 mm and produces a field of 0.73 T, how many “Ampere-turns” is required of the magnet design?

  2. Now introduce a slope to the poles, creating a gradient magnet, similar to that of the Fermilab Booster magnet shown in class. Derive an estimate for the field gradient \(\partial B_y/\partial x\) if the two pole faces have \(\pm 10^\circ\) degree slopes (resulting in a positive gradient).

  3. Using FEMM, create a realistic model of the above gradient magnet. Compare the value of the field gradient to that of your estimate above.

  4. The Fermilab Booster BF magnet uses 24 turns of conductor per pole. What current is required to generate the field needed for an 8 GeV (kinetic energy) proton beam? Assume the gap height on the ideal orbit is 50.8 mm. [Note: See the Booster parameters found in Problem Set 3.]

  5. Using your FEMM model as an approximation to the BF magnet, what slope of the pole tips is necessary to generate the gradient required for the Fermilab BF magnet? [Again, see the Booster parameters found in Problem Set 3 to find the necessary gradient.]

MRI Solenoid Magnet

A solenoid magnet to be used for an MRI instrument requires a coil with inner bore radius 0.7 m and a central field strength of 1.5 T.

  1. Assume the MRI magnet coil is a classical solenoid with \(N'\) turns of conductor per meter in the \(z\)-direction and of length much greater than the bore radius. Calculate the necessary value of \(N'\times I\) to produce the desired field value.

  2. If we again assume zero thickness for the coil, but finite length of \(L\), then one can show that the magnetic field on axis will be \[ B_z(z) = \frac{\mu_0 N'I}{2} \left( \frac{z+L/2}{\sqrt{R^2+(z+L/2)^2}} - \frac{z-L/2}{\sqrt{R^2+(z-L/2)^2} } \right) . \] Calculate and plot the magnitude of the field on the central axis as a function of \(z\) over the range -3 m < \(z\) < 3 m for a solenoid of length 2.2 m.

  3. Perform a FEMM calculation for a solenoid magnet with the above properties. For this calculation, assume a coil thickness of 14 mm (i.e., 2% of the coil inner radius). Make a single plot of both the FEMM results of \(B_z(z)\) along with the curve found in part b.

[Note: FEMM ships with a magnetic-solenoid example, “Magnetics Tutorial,” which should be in the directory C:femm42\examples\tutorial.fem on the user’s local machine. Many other useful examples can be found at the FEMM home page, under “Examples”.]

  1. (Extra Credit): Derive the equation in part b above.