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    MHD problem

    Homework Statement Two of the MHD equations can be formulated as \vec{E} + \vec{v} \times \vec{B} = \eta \vec{J} \nabla \times \vec{B} = \mu_0 \vec{J} where [itex]\eta[/tex] is the resistivity of the plasma. a.) Derive an equation for the magnetic field at very high resistivity and...
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    Show that a longitudinal wave is electrostatic

    Homework Statement Show that all longitudinal waves must be electrostatic by using Faraday's law. Homework Equations Faraday's law: \frac{\partial \vec{B}}{\partial t} = - \nabla \times \vec{E} The Attempt at a Solution Where should I start??
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    Solar wind at jupiter

    Homework Statement I'm trying to solve a problem related to the solar wind pressure at jupiter but I'm stuck at calculating the density. It is stated that the solar wind has a density of 5 [itex]cm^{-3}[/tex] and a speed of 400 km/s at the orbit of the Earth, and that it should be assumed...
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    Plasma MHD

    Homework Statement Two of the MHD equations can be written as \vec{E} + \vec{v} \times \vec{B} = \eta \vec{J} \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} where [itex]\eta[/tex] is the resistivity of the plasma. Derive an expression for the magnetic field at a very high resistivity...
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    Plasma physics

    Homework Statement Derive (from the equation of motion of a neutral gas and an assumption of constant gravitational field) an expression showing why the concentrations of neutral molecules decrease approximately exponentially with increasing altitude, and why the concentration of atomic oxygen...
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    Complex integration

    How do I solve an integral of the type \int f(v) e^{iavx} dv ? Can I just treat i as any other constant?
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    Rayleighs per count for a CCD

    How can I find the number of Rayleighs per count if I know the column emission rate, radiance, irradiance, #photons per pixel and #photoelectrons per pixel? I'm totally lost in this one, please help!
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    Time varying heat conduction

    Hey everyone! I am currently on a project building a small CanSat. This is a small satellite of the size of a coke can which will be launched together with a balloon and then descend from an altitude of 35 000 m. My problem now is to work out the heat conduction to see if our insulation is...
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    Wavelength of particle motion

    I have two parametric equations for the speed of a particle in a plane: \dot{x}(t) = A \left( 1 - cos{\Omega t} \right) \dot{y}(t) = A sin{\Omega t} The period is equal to \Omega. How do I find the wavelength of the motion? The wavelength is just \lambda = \Omega v , where v =...
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    Determine the force in z-direction on the gyrocenter of a charged particle

    Homework Statement Determine the force in z-direction on the gyrocenter of a charged particle in a diverging magnetic field. \frac{\partial B}{\partial z} < 0 Homework Equations The Attempt at a Solution Please give me a starter. Could I use the Lorentz force in this case?
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    Orbit calculations

    Homework Statement The given data is the perigee altitude r_p, the apogee altitude r_a and the period T. Mission: find the altitude 30 min after perigee passage. Homework Equations Semi-major axis a is calculated. Kepler's equation gives a relation for the eccentric anomaly E: E -...
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    Root loci

    Homework Statement For the double integrator described with transfer function G(s) = \frac{1}{s^2} the initial condition is zero. The double integrator is subjected to a unit‐feedback system where the controller is chosen as 1) a PI-controller with C(s) = k_p \left( 1 + \frac{1}{s}...
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    Control theory

    Homework Statement I have a transfer function G(s) = \frac{1}{s^2} and a PI controller P(s) = 6 \left( 1 + \frac{1}{s} \right). How do I check for stability? Just use 1 + P(s)G(s) = 0 and check the roots?
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    Gauge transformations in GR

    I have been told that using a metric g_{00} = -a^2(\eta)(1+2\psi) g_{oi} = g_{i0} = a^2(\eta)\omega_i g_{ij} = a^2(\eta) \left[(1+2\phi)\gamma_{ij} + 2\chi_{ij} \right] and a gauge transformation x^{\bar{\mu}} = x^{\mu} + \xi^{\mu} with \xi^0 = \alpha \xi^i = \beta^j gives...
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    Irrotational field -> Symmetric Jacobian

    Does anyone know any reference or proof to the statement that since a flow is irrotational, the Jacobian is symmetric?
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    Inflationatory universe

    Homework Statement This problem concerns a simplified model of the history of a flat universe involving a period of inflation. The history is split into four periods: (a) 0<t<t_3 radiation only (b) t_3<t<t_2 vacuum energy dominates, with an effective cosmological constant \Lambda =...
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    Luminosity integral help

    If the total luminosity is given by L_T = \int_0^{\infty} e^{-(r/a)^{1/4}} r^2 dr estimate the radius [tex]r(a)[/itex] corresponding to half of the total luminosity. This would be to integrate from zero to r and get a function r(a,L) but this is impossible so anyone got an idea on how...
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    Differential equation

    Can anyone help me solve \dot{x} = Hx_0 \left( \sqrt{B} \frac{x_0}{x} + \sqrt{1-A-B} \right)
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    Vacuum energy density

    Homework Statement Show that, in natural units h=c=1, an energy density may be expressed as the fourth power of a mass. If the vacuum energy contributed by a cosmological constant is now of order of the critical density, what is the mass to which this density corresponds? 2. The attempt...
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    Statistical physics

    Homework Statement The result n_{0 \gamma} = \left( \frac{k_BT_{0r}}{hc} \right)^3 \int_0^{\infty} \frac{8 \pi x^2 dx}{e^x-1} = 2 \frac{\zeta(3)}{\pi^2} \left( \frac{k_BT_{0r}}{hc} \right)^3 is obtained for photons by integrating over the Planck distribution appropriate for bosons. In...
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    Perfect square

    Homework Statement \left(\frac{\dot{x}}{a}\right)^2 = K\left[b\frac{a}{x} + c\left(\frac{a}{x}\right)^2 + (1-b-c)\right] Show that, for b<1, there is a value of c that makes the right hand side a perfect square of a function of x. 2. The attempt at a solution I guess that a perfect...
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    Nucleocosmochronology: hydrogen/helium ratio and its change

    Homework Statement Assume that the mass-to-light ratio, M/L, for the galaxy is, and has always been, 10 in solar units. What is the maximum fraction of the total mass that could have been burnt into helium from hydrogen over 10^{10} years? (The mass deficit for the reaction 4H \rightarrow ^4He...
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    Surface brightness

    Homework Statement Show that the Hubble profile of surface brightness I(r) = I_0 \left(1+\frac{r}{R}\right)^{-2} leads to an infinite total luminosity, while the law I = I_0 exp[-(r/a)^{1/4}], with a a constant, does not. Here I_0 and R are constants and r is the distance from the centre...
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    Singularities and Kasner solution

    Homework Statement Investigate the possible behaviour of the singularity as t \rightarrow 0 in the Kasner solution. Homework Equations The metric for the Kasner solution is given by ds^2 = c^2dt^2 - X_1^2(t)dx_1^2 - X_2^2(t)dx_2^2 - X_3^2(t)dx_3^2 The Attempt at a Solution I...
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    Rotation curve

    How can I calculate the rotation curve, v(R), for test particles in circular orbits of radius R around a point mass M?
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    Luminosity physics question

    Homework Statement Show that the Hubble profile of surface brightness I(r) = I_0 \left(1+\frac{r}{R}\right)^{-2} leads to an infinite total luminosity. r is the distance from the center and R is a constant. 2. The attempt at a solution For large r this is related as I(r) \propto...
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    Coordinate transformation

    How can I identify the coordinate transformation that turns ds^2 = \left(1+\frac{\epsilon}{1+c^2t^2}\right)^2c^2dt^2 - \left(\frac{\epsilon}{1+x^2}\right)^2x^2 - \left(\frac{\epsilon}{1+y^2}\right)^2y^2 - \left(\frac{\epsilon}{1+z^2}\right)^2z^2 into the Minkowski metric ds^2 = c^2dt^2...
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    Conformal time closed Friedmann universe

    Homework Statement A closed Friedmann universe contains a single perfect fluid with an equation of state of the form p=w\rho c^2. Transforming variables to conformal time \tau using dt=a(t)d\tau, show that the variable y=a^{(1+3w)/2} is described by a simple harmonic equation as a function of...
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    Friedmann Equation

    Homework Statement By substituting in \left( \frac{\dot{a}}{a_0} \right)^2 = H^2_0 \left(\Omega_0 \frac{a_0}{a} + 1 - \Omega_0 \right) show that the parametric open solution given by a(\psi)=a_0 \frac{\Omega_0}{2(1-\Omega_0)}(\cosh{\psi} - 1) and t(\psi)=\frac{1}{2H_0} \frac{\Omega_0}{(1 -...
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    Coordinate distance

    Homework Statement For a universe with k=0 and in which (a/a_0) = (t/t_0)^n where n<1, show that the coordinate distance of an object seen at redshift z is r=\frac{ct_0}{(1-n)a_0}[1-(1+z)^{1-1/n}]. 2. The attempt at a solution I have used r=f(r)=\int_{t}^{t_0}...
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