There is no net field inside the sphere and so the field inside the sphere due to only the shell is just the opposite to the field due to q1 (the charge of the shell is irrelevant, this equal and opposite field would be created whether the sphere had charge or not).
The question, however, asks...
Cleonis' post provides remarkable insight to this North/South motion, and I would suggest that the equator exists as a choice of co-ordinate system (polar in this case), not the objects rotation.
As for the restricted plane comment, you can't forget that you have modeled an inertial reference...
You've worked out from Gauss' law that the net force on the charge at the centre should be zero. Now you can say that the net force on this charge is actually due to a contribution from the shell and the point charge to the right... hope this helps?
Good good, that's what I was about to say, the centrifugal force is modeled and doesn't need to be added. OK, that just leaves the sign of the vector product under question, I still think that your accelerations from the vector product term should have a minus (I previously thought the...
Ok so, in rotating reference frame:
\frac{\rm{d}^2\mathbf{r'}}{\rm{d}t^2} = \frac{\rm{d}}{\rm{d}t}\left(\dot{\mathbf{r}} + \omega\times\mathbf{r}\right)
So expanding this out I get...
\frac{\rm{d}^2\mathbf{r'}}{\rm{d}t^2} = \ddot{\mathbf{r}} + \mathbf{\omega}\times\dot{\mathbf{r}} +...
OK, I want to confirm that your vector multiplication for the acceleration was:
\ddot{\mathbf{r}} = - \dot{\mathbf{r}} \times \mathbf{\omega}
In which case only your a_y has the correct sign. I was curious as to why you put a factor of 2 in front of these??
Also, there is no need for...
Exactly my point, the fact that sin(x)/x for a particular value of x may not be defined can be misleading. Especially when, in reality, we usually deal with continuous variations where undefined values suddenly are defined!??
Apologies lifelearner, we're digressing from the problem.
One final question... the most important one I've asked... have you modeled the Earth's rotation? If you have then I agree with the program for reasons I will disclose.
And what are your units here (how high does the projectile reach)? If you have neglected the variation of height with gravity then maybe this assumption is not valid and you are getting the Northern deflection because of a computational reason.
Can you post your code?? (commented if poss).
Sorry, missed the j term in the exponential with \beta_2. But I still think it doesn't belong in the expression for \alpha_2. It corresponds to the propagation constant in the x direction.
The first part can be done by just considering that the field inside the medium can be represented as a sum of cosine/sine waves corresponding to the reflected and incident wave. The reflected ray will have experienced a phase delay upon reflection according to the Frensel equations. So the...
Does 2*9.8*10.1 really equal 203.84?? (I can see without using a calculator that this number is not going to exceed 200)
And in your flow rate equation, the radius should be squared. I think your equations are fine, but not the "maths".
I think this question is a bad question for those learning maths, the sinc function for example (\rm{sinc}\,(x) = \frac{sin x}{x}) at x = 0 becomes:
\rm{sinc}\, 0 = \frac{0}{0} = 1
but this value is certainly well-defined (obtainable with L'Hopital's rule). I think the quantity zero needs...