Question

a) Show that the real and imaginary parts of (2.41) agree with the first two solutions in (2.33).

eq 2.41:- μ+(t) = μ+(0)e−iω0t

eq 2.33:- μx(t) = μx(0) cos ω0t + μy(0) sin ω0t μy(t) = μy(0) cos ω0t − μx(0) sin ω0t μz(t) = μz(0)

textbook: Magnetic resonance imaging, physical principles and sequence design

b) Demonstrate that (2.24) implies dμ/dt = 0. Hint: Form a scalar (dot) product
of both sides of (2.24) with ~μ.

eq 2.24: dμ/dt = γμ × B

Answer 1

a) given

Substitute

and

Hence

Separate the real and imaginary parts on the right side

or

Equate the real and imaginary parts on both sides

and

which are equations 2.33

b) given

take dot product with mu on both sides

The right hand side is zero because

and cross product of any vector with itself is zero

Therefore

Or

or

or