If the differential size-frequency distribu- tion of a group of objects that have equal densities can be adequately described by a power law in radius of the form N(R)dR ∞ R-3, then it can also be described as a power law in mass of the form N(m)dm ∞ m¯*. Derive the relationship between g and x.
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- The energy E of a system of three independent harmonic oscillators is given by 1 E = (Nx + ½)ħw + (ny + ½ )ħw + (nz + 1⁄)ħw. 2 (20.49) Show that the partition function Z is given by Z = ZSHO, (20.50) where ZSHO is the partition function of a simple harmonic oscillator given in eqn 20.3. Hence show that the Helmholtz function is given by 3 F = = ħw + 3k³Tln(1 – e¯ßhw), (20.51) and that the heat capacity tends to 3kB at high temperature.Find the minimum acceleration needed for a maglev train to reach a speed of 430 km/h in 3.0 km, one tenth of the length of the 30-km track in Shanghai. Express your answer as a fraction of g, the free-fall acceleration near Earth. Would this acceleration be noticeableIn the upper half space, which is the empty space, I = 7 A current flows from the infinitely long wire along the y axis that intersects the z axis at the point C (0,0,10). Half-space z <0 is from a material with relative magnetic permeability µr = 5. Write numerically the sum of the components (Hx + Hy + Hz) at the point A (-5, -5,0 +) of the magnetic field H [A / m] vector in terms of the given magnitudes (in the half space z> 0).
- Please asapThe motion of electrons in a conductor is purely random in nature as the electrons do not follow a strict path to move from one location to the other. For a short burst of time, one of the electrons in a conductor being studied is identified to be moving approximately in the following manner: (a) (b) 7=(cos(zy)-x, cos(xz) — y, cos(xy) - z). Calculate the divergence of 7. Determine whether the electron has any tendency to rotate at point (0,0,0).Evaluate the equation of continuity of plane wave function Exp i (kx-wt).
- In quantum mechanics, the probability of finding a particle æ in domain (a, b) can be calculated from the following formula P(a < # < b) = / Mla p(x)dr where the probability density p(a') satisfies p(x) « 1/x for æ e (1, e²) and being 0 elsewhere. Please determine the exact form of the probability density and calculate the probability of finding a particle in domain [1, 2).Consider an clectron in a uniform magnetic field pointing along the z direction. The electrou spin is measured (at time to) to be pointing along the positive y-axis. What is the polarization along the x and z directions (i.e. the expectation values of 2sz and 2sz) for t > to?(a) Use the standard definition of the average value of a random variable given its probability density function: = 10 x p₁(x) -∞ (9) to find average position of the wave-particle for each stationary state. Recall that pn = Un is the probability density associated with the stationary state n = 1,2,... for the particle in a box. Also, n(x, t) is only defined for x = (0, L). Hint: the result is the same for all stationary states. That is, it does not depend on n. n= dx.