Question

29C

MRI Magnetic Resonance Imaging takes exquisite images of the brain and other parts of human body. The magnetic field B is generated by a solenoid of length 3 feet and diameter of 3 feet. The B field generated is one (1) T. The largest current the wire can carry is 100 A.

a) How many turns of solenoid is needed? Eq. (29.17)

b) Find the total length of solenoid wire in miles. A proton (hydrogen nucleus) has nuclear magnetic moment μ = 8.8e-8 eV/T. It has two energy states ± μ∙B. Transitions between the two energy states give radiation detected by radiofrequency detector. For a 1-T magnetic field, find the:

c) energy difference in neV (e-9 eV) between the two states.

d) frequency of this signal. (This is well known in MRI.) Eq. (38.4)

Answer 1

a)
Magnetic field of a solenoid, B = (o * N * I) / L
Where o is the vacuum permeability, N is the number of turns, I is the current and L is the length of the solenoid.
N = (B * L) / (o * I)
Substituting values,
N = (1 * 0.914) / (4 * 10^-7 * 100)
= 7.28 * 10³

b)
Total length of the wire, S = N * 2 * R
Where R is the radius of the solenoid.
Substituting values,
S = (7.28 * 10³) * (2 * 0.4572)
= 20.9 * 10³ m
= 20.9 km
= 13 miles

c)
Energy, E = 2 * * B
Where is the magnetic moment and B is the magnetic field.
E = 2 * (8.8 * 10^-8) * 1
= 17.6 * 10^-8 eV
= 176 * 10^-9 eV
= 176 neV

d)
E = h *
Where h is the Plank's constant and is the frequency
E = 176 * 10^-9 eV
= 176 * 10^-9 * 1.6 * 10^-19 J
= 2.82 * 10^-26 J

= E / h
= (2.82 * 10^-26) / (6.626 * 10^-34)
= 4.25 * 10⁷ Hz