Question

4. Let β = Nφ , show that as N-> infinite and φ → 0 The intensity → I = 4 Im * ( sin(β/2) / β )^2 Where Im = N^2 * I0

a) show that minimum is at a*sin(θ ) = m λ and maximum is at a*sin(θ ) = (m+1/2) λ (Intensity drops drastically) Where a is the width of a single slit

b)Show that if I: intensity inversely square decreases of r , distance from the source, then he amplitude inversely decreases of r.

Answer #1

(8.0.10) In class we showed that intensity distribution for
single slit diffraction is given by I(θ) = I(0) [sin( β/ 2)/(
β/2)]. ( β = ka sin(theta) , "a" is the slit width and k = 2π/λ ).
(a) Show that the maxima in the diffraction pattern occur at θ for
which tan( β/ 2) = β/2. (b) An obvious solution to this equation is
β = 0, which corresponds to the central maximum. By sketching the
functions tan(β/2) and...

1. Given β = XT 1×nAn×nXn×1, show that the gradient of β with
respect to X has the following form: ∇β = X T (A + A T ). Also,
simplify the above result when A is symmetric. (Hint: β can be
written as Pn j=1 Pn i=1 aijxixj ).
2. In this problem, we consider a probabilistic view of linear
regression y (i) = θ T x (i)+ (i) , i = 1, . . . , n, which...

(A) Find an impedance, Z, of an RLC ciruits containing R = 2 , L =
1 mH, and C = 1 F elements connected in series with with the
voltage source E = E0 sin( Dt), where D is the driving frequency.
(B) Find the expression for the current I. Plot the current
amplitude I0 vs. D and show that it is maximum for D = 1/(LC). We
call this maximum a resonance (C) Consider the case with no...

(4) e/m = v/Br
(5) B=(N)(I)8μ0/R(√125) , N is the number of turns on each coil,
I is the current, R is the radius, and μ0 = 4pi x 10-7 Tm/A.
(8) v = (2eV/m)^1/2
Using Equations 4, 5, and 8, solve for e/m in terms of r (radius
of the path), R (average radius of the coils), V (voltage of the
electron source), I (current in the coils), N (number of turns of
the coils) and constants. Hint: start...

6) (8 pts, 4 pts each) State the order of each ODE, then
classify each of them as
linear/nonlinear, homogeneous/inhomogeneous, and
autonomous/nonautonomous.
A) Unforced Pendulum: θ′′ + γ θ′ + ω^2sin θ = 0
B) Simple RLC Circuit with a 9V Battery: Lq′′ + Rq′ +(1/c)q = 9
7) (8 pts) Find all critical points for the given DE, draw a phase
line for the system,
then state the stability of each critical point.
Logistic Equation: y′ = ry(1 −...

1. Consider the Markov chain {Xn|n ≥ 0} associated with
Gambler’s ruin with m = 3. Find the probability of ruin given X0 =
i ∈ {0, 1, 2, 3}
2 Let {Xn|n ≥ 0} be a simple random walk on an undirected graph
(V, E) where V = {1, 2, 3, 4, 5, 6, 7} and E = {{1, 2}, {1, 3}, {1,
6}, {2, 4}, {4, 6}, {3, 5}, {5, 7}}. Let X0 ∼ µ0 where µ0({i}) =...

Question 4: Match the words (from the options below) to the
statements by inserting the corresponding capital letter in the
blank. No word is used more than once (2 Points for (i) to (xii), 1
point for (xiii).
(i) If you double the frequency of electromagnetic radiation you
______ the energy.
(ii) If you double the wavelength of electromagnetic radiation
you _____ the energy.
(iii) A ____ produces an electric signal when struck by
photons.
(iv) A ____ produces many...

II(20pts). Short Problems
a) The lowest energy of a particle in an infinite one-dimensional
potential well is 4.0 eV. If the width of the well is doubled, what
is its lowest energy?
b) Find the distance of closest approach of a 16.0-Mev alpha
particle incident on a gold foil.
c) The transition from the first excited state to the ground
state in potassium results in the emission of a photon with = 310
nm. If the potassium vapor is...

1) State the main difference between an ODE and a PDE?
2) Name two of the three archetypal PDEs?
3) Write the equation used to compute the Wronskian for two
differentiable
functions, y1 and y2.
4) What can you conclude about two differentiable functions, y1 and
y2, if their
Wronskian is nonzero?
5) (2 pts) If two functions, y1 and y2, solve a 2nd order DE, what
does the Principle of
Superposition guarantee?
6) (8 pts, 4 pts each) State...

Please answer all not understanding:
1. Volumetric expansion coefficients of simple materials are
often well catalogued. However, the thermal expansion coefficient β
of a human body is less well known. This could affect the human
body\'s specific gravity and, therefore, measurements of its
body/fat ratio. Suppose that a human body of weight w0 (on dry
land) is placed on a scale while completely immersed in
formaldehyde of temperature T1. Once the temperature increases by
ΔT, the scale reading drops by...

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