Problem Assignment 1
1. Consider the ratio of M 0 T1  M 0 T2  , where T1 and T2 are temperatures. Give a formula for
the ratio in terms of the static magnetic field, B0, the two temperatures, T1 and T2, the
gyromagnetic ratio for protons, Boltzman’s constant, Planck’s constant, and any other
parameters you need.
2. Using your formula for M 0 T1  M 0 T2  , calculate its numerical value when B0 = 1.5Tesla,
T1.= 70 degrees Fahrenheit, and T2 = 40 degrees Fahrenheit.
3. Give the limit of your formula for M 0 T1  M 0 T2  , (a) as T1 approaches zero and T2 remains
finite, and (b) as T2 approaches infinity and T1 remains finite.
4. The Bloch equation is a vector equation that gives the evolution of all three components of
the magnetization, Mx, My, and Mz. Give three equations for the evolution of the
components—one each for Mx, My, and Mz. These three equations are called the “Bloch
equations” (i.e., plural form). More specifically, they are the Bloch equations “in the
laboratory frame”.
5. Hornak, Ch. 3, Problem 4. Hornak gives the answer. Don’t turn it in. Just make sure you can
do it!
6. When B  B0 zˆ and Mz starts at zero and begins relaxing toward its equilibrium value, as
dictated by the Bloch equations, it obeys the formula,
M z  M 0 1  e  t / T 1  .
Give the formula when Mz has some arbitrary initial value Mi (i.e., the subscript i means
“initial”). (b) Give the specific formula for the time immediately following a 180-degree RF
pulse (t = 0 at the moment that the pulse is turned off).
7. Suppose M has reached its equilibrium value, and an RF pulse is applied for a time  .
Suppose also that in the rotating frame B1 is directed along the x axis and both T1 and T2
can be considered to be infinitely large (i.e., in comparison to  ). Write the Bloch equations
in the rotating frame for this case and show that they are solved by
M y  M 0 sin   B1t  ,
M z  M 0 cos   B1t  ,
where t = 0 at the beginning of the RF pulse. Show it by plugging these formulas into the
two sides of the appropriate equations and showing that the results are the same for the
respective sides.
8. Suppose that the static magnetic field has magnitude B and is pointed in the z direction (as
usual), and suppose that M = Ma ẑ . (note that the subscript is an “a”, meaning “arbitrary”,
not a zero). Now suppose that a 90-degree RF pulse is applied. Let time t=0 at the moment
that the pulse is terminated. Starting at t=0, the following three equations solve the three
Bloch equations that you derived above for the laboratory frame:
M x  c1 sin  c0 t  e  c2t ,
M y  c3 cos  c0 t  e  c2t ,


M z  c4 1  e  c5t ,
if the correct constants, c0 through c5, are used. Give those constants.
9. Hornak, Ch. 3, Problem 5. Hornak gives the answer. Don’t turn it in. Just make sure you can
do it!
10. Hornak, Ch 3, Problem 6. Hornak gives the answer. Don’t turn it in. Just make sure you can
do it!