I Gravitational Illusions -- Two balls rolling down differently-shaped ramps

AI Thread Summary
The discussion centers on the concept of gravitational illusions, specifically how two balls roll down differently-shaped ramps. Participants emphasize the importance of hands-on problem-solving to truly understand the physics involved, rather than relying solely on mathematical proofs. The Brachistochrone curve is mentioned as a relevant mathematical solution to the problem of determining which ball reaches the bottom first. Contributors are encouraged to share their own mathematical solutions for feedback and discussion. Engaging with the problem directly is highlighted as essential for deeper learning.
Jurgen M
Without this test,using only math, would you prove that left ball will comes first?

 
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Why make us do all the work? You'll never learn unless you do the problems yourself.

Write your own mathematical solution and post it here. We'll comment on it.
 
Thread 'Gauss' law seems to imply instantaneous electric field'

Thread 'Gauss' law seems to imply instantaneous electric field'

Imagine a charged sphere at the origin connected through an open switch to a vertical grounded wire. We wish to find an expression for the horizontal component of the electric field at a distance ##\mathbf{r}## from the sphere as it discharges. By using the Lorenz gauge condition: $$\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t}=0\tag{1}$$ we find the following retarded solutions to the Maxwell equations If we assume that...
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Thread 'Off resonance driving of a MW/RF cavity'

Hello! Let's say I have a cavity resonant at 10 GHz with a Q factor of 1000. Given the Lorentzian shape of the cavity, I can also drive the cavity at, say 100 MHz. Of course the response will be very very weak, but non-zero given that the Loretzian shape never really reaches zero. I am trying to understand how are the magnetic and electric field distributions of the field at 100 MHz relative to the ones at 10 GHz? In particular, if inside the cavity I have some structure, such as 2 plates...
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