Consider a particle of a potential energy V(x) = {î where 1 is a positive constant....
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
Earth ny In. 2. Using Lagrange's equation, write the equations of motion of the spacecraft (particle Q) in problem 4. The potential energy of a particle in Earth's central gravity field is: V The negative sign arises because the gravity potential is defined as zero at r-o The resulting equations of motion should be the same as those in problem 4. G M m
Earth ny In.
2. Using Lagrange's equation, write the equations of motion of the spacecraft (particle...
E70.1(a) Construct the potential energy operator of a particle with potential energy V(x)={kx, where k is a constant.
#1 A particle of mass, m, moves in a field whose potential energy in spherical coordinates has a 2 , where r and are the standard variables of spherical coordinates and k is a positive constant. Find Hamiltonian and Hamilton's equations of motion for this particle. form of V --k cose
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4. Consider a classical particle at temperature T. Suppose the Hamilton (i.e. the total energy) function H for the particle can be written as a sum of independent quadratic terms in the variables on which H depends. That is, if H -H(31,£2... ), then Here,5 could be a position or a momentum coordinate, and the a's are constants. As an example, 2 2 Px for a ID...
1. A particle is scattered upon by the potential V(x)-re(x). (β is a positive constant) (a) Write down the proper boundary conditions. (b) Find the reflection and transmission coefficients for E>0. (20%)
A particle of mass m is in a potential energy field described by, V(x, y) = 18kx² +8ky? where k is a positive constant. Initially the particle is resting at the origin (0,0). At time t = 0 the particle receives a kick that imparts to it an initial velocity (vo, 2vo). (a) Find the position of the particle as a function of time, x(t) and y(t). (b) Plot the trajectory for this motion (Lissajous figure) using Vo = 1,...
A particle with positive charge q = 1.12 10-18 C moves with a velocity v with arrow = (5î + 2ĵ − k) m/s through a region where both a uniform magnetic field and a uniform electric field exist. (a) Calculate the total force on the moving particle, taking B with arrow = (5î + 2ĵ + k) T and E with arrow = (2î − ĵ − 4k) V/m. (b) What angle does the force vector make with the...
A particle with positive charge q = 4.01 10-18 C moves with a velocity
v
= (5î + 5ĵ − ) m/s
through a region where both a uniform magnetic field and a uniform
electric field exist.
(a)
Calculate the total force on the moving particle, taking
B
= (5î + 2ĵ + ) T
and
E
= (3î − ĵ − 2) V/m.
(Give your answers in N for each component.)
Fx= NFy=
NFz= N
(b)
What angle does...
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struggling!
A Cycloidal Path A particle with mass m and positive charge q starts from rest at the origin as shown in the figure below. There is a uniform electric field = Eo? and a uniform magnetic field B = BoZ The path of the charged sketched below is a cycloid whose radius of curvature at the top points is twice the y-coordinate at that level. This path is exactly the same as the path of...