I am studying for a comprehensive exam on dynamic contrast enhanced MRI (DCE-MRI) and questions have come up in my practice talks about dynamic susceptibility contrast MRI (DSC-MRI).
In DCE-MRI changes in signal in an artery and the tissue are converted to contrast agent concentration vs time...
Consider a spin system with noninteracting spin 1/2 particles. The magnetic moment of the system is written as:
μ = (ħq/2mc)σ
Where σ = (σx, σy, σz) is the Pauli spin operator of the particle. A magnetic field of strength Bz is applied along the z direction and a second...
Consider a particle with mass m oscillates in a simple harmonic potential with frequency ω. The position, x, and momentum operator, p, of the particle can be expressed in terms of the annihilation and creation operator (a and a† respectively):
x = (ħ/2mω)^0.5 * (a† + a)
The Hamiltonian of an electron in solids is given by H. We know that H is an Hermitian operator, it satisfies the following eigenvalue equation:
H|Φn> = εn|Φn>
Let us define the following operators in terms of H as:
U = e^[(iHt)/ħ] , S = sin[(Ht)/ħ] , G = (ε -...
Consider A(x) is an arbitrary function of x, and px is the momentum operator. Show that they satisfy the following condition:
[px,A(x)] = (-i/ħ)*d/dx(A(x))
where [px,A(x)] = pxA(x) - A(x)px
ħ = h/2π
px = (-iħ)d/dx
The Attempt at a Solution
We have an infinite slab of conducting material, parallel to the xy plane, between z = −a and z = +a, with magnetic susceptibility χm. It carries a free current with volume current density J = J0z/a in the x direction (positive for z > 0, negative for z < 0). The integrated...
A square wire loop of size 2a by 2a lies in the x- y plane with its center at the origin and sides parallel to the x and y axes. A counterclockwise current I runs around the loop.
(a) Find the magnetic field on the z axis. [Answer: Bz = 2μ0Ia2/[(a2 + z2)(2a2 + z2)0.5]]
A long, straight wire, radius R, carries total current I. The current is distributed in the wire so that the current
density is proportional to s, the distance from the center of the wire.
(a) Write an expression for the current density J in the wire, as a function of s...
I need to calculate the electric field on the midplane of a uniformly polarized cylinder at a large distance from the center of the cylinder. The question also says that because the distance is large compared to the radius the dipole dominates the multipole expansion...
2 concentric conducting spherical shells, with radii a and 2a, have charge +Q and -Q respectively. The space between the shells is filled with a linear dielectric with permittivity ε(r) = (ε0*a)/(1.5*a - 0.5*r), which varies with distance r.
a) Use Gauss's law to determine...
I'm doing a question that requires me to take the dot product of 2 vectors in spherical coordinates. Both vectors have only an r component, can I just multiply the r components?
The Attempt at a Solution
Consider a uniformly polarized electret in the shape of a cylinder with height h and radius 10h. The polarization in the dielectric is P, parallel to the cylinder axis.
a) calculate surface and volume bound charges
b) calculate the electric field at the center of the...