Recent content by GL_Black_Hole

Homework Statement A rigid cylinder of radius ##R## and mass ##\mu## has a moment of inertia ##I## around an axis going through the center of mass and parallel to the central axis of the cylinder. The cylinder is homogeneous along its central axis, but not in the radial and angular directions...

So then the rest of the solution would proceed `as I've outlined? Use the expression for curl in spherical coordinates, find which component of B is non-zero and then turn the crank to find Poynting's vector and compare with the time derivative of the electrostatic energy? Regarding units I'm...

Homework Statement A spherical capacitor has internal radius ##a## and external radius ##b##. At time ##t = 0##, the charge of the capacitor is ##Q_0## Then the two shells are connected by a resistor in the radial direction of resistance ##R##. Find the Poynting vector and the energy...

Yes that is what I meant by time in the lab frame. Thank you for reinforcing the distinction between the time the clock shows and the time in the lab frame for me.

Possibly. It doesn't directly mention a reference frame. But even so am I correct in how I'm finding the time in the lab frame? Or is there a way to use the Lorentz transformations more directly?

Homework Statement A rectangular structure carries clocks at its four corners. The clocks are synchronized in the structure’s rest frame, in which it has length L =4ft and width W = 3ft. In our laboratory frame the rectangle is moving in the positive x direction at speed v = 0.8c. As the clock...

Homework Statement Show that it is impossible for an ultra-relativistic particle with ##pc>>Mc^2## to disintegrate into two identical massive particles of mass m. Homework Equations Conservation of four momentum The Attempt at a Solution The four momentum of the ultra-relativistic particle...

Homework Statement Consider an inertial laboratory frame S with coordinates (##\lambda##; ##x##). The Lagrangian for the relativistic harmonic oscillator in that frame is given by ##L =-mc\sqrt{\dot x^{\mu} \dot x_{\mu}} -\frac {1}{2} k(\Delta x)^2 \frac{\dot x^{0}}{c}## where ##x^0...

Homework Statement Derive the relativistic Euler equation by contracting the conservation law $$\partial _\mu {T^{\mu \nu}} =0$$ with the projection tensor $${P^{\sigma}}_\nu = {\delta^{\sigma}}_\nu + U^{\sigma} U_{\nu}$$ for a perfect fluid. Homework Equations $$\partial _\mu {T^{\mu \nu}} =...

Homework Statement The surface tension of a layer of water obeys ## \sigma = a- bT##, where ##T## is the temperature. Find the change in temperaure, ##\Delta T## when the area is increased isentropically. Homework Equations ## dU = dQ -dW## , ##dW = \sigma dA##, ##dU = C_A dT + [\sigma...

Homework Statement The coefficient of thermal expansion and isothermal compressibility of a gas are given by ##\alpha_P =\frac{V-b}{TV}## and ##\kappa_T = \frac{V-b}{PV}## find: a) The equation of state b) If the heat capacity at constant volume ##C_V## is constant, what is ##\delta U##? c)...

Homework Statement Show that ##(\frac{\partial S}{\partial G})_Y = -\frac{C_Y}{TS}## Homework Equations ##G = H-TS, (\frac{\partial H}{\partial T})_Y = C_Y## The Attempt at a Solution ##dG = dH -TdS -SdT## and ##H## is a state variable so ## dH =\frac{\partial H}{\partial T} dT +...

So I've worked out the rest of the question and I can get the sensible answer that the rate of energy dissipation per unit volume, ## J \cdot E##, equals the rate of decrease of the electrostatic energy of the capacitor, ## - \frac{d}{dt} [\frac{1}{2} E \cdot D]##, if the Joule heating term only...

Right, electric displacement is used for dielectrics. Correcting that error gives me: ## E = \frac{-\lambda_{free}}{2\pi \epsilon r}##, so the DE for the charge density becomes ##\frac{d}{dt} \lambda_{free} = -\frac{\sigma}{\epsilon} \lambda_{free}##, and the displacement current with the...

Thank you for the reply. I've attempted the question up to the energy dissipation part now. The problem stated that the central conductor has ##-\lambda_{free}##, so applying Gauss' Law to a cylinder contained in the dielectric at a radius r and with a length L the electric field inside the...