As a side note (and I'm not sure if this is appropriate to chat about here, so please feel free to move/tag this as needed), this class on electromagnetic wave theory is a bit of a challenge for me, in that I find myself having difficulty choosing the appropriate "strategies" of knowing when to...
Not sure if my browser is at fault, but the LaTex should be there in the post. I did just edit my response; maybe it was a syncing issue 🤔 In any case, the "Therefore" is:
$$
\begin{align} \nonumber
\frac{\rho}{\varepsilon_{0}} & = \frac{2E_{0}}{r} \\ \nonumber
\rho & = \frac{2 \varepsilon_{0}...
Yes, sorry about that. I'll try to keep these things sorted better.
I think these replies have helped me to get the same answer in two different ways:
Method 1:
$$
\begin{align} \nonumber
E_{0} \ 4 \pi \varepsilon_{0} \ r^{2} & = \int_{0}^{r}{\rho(r) \ r^{2} \ dr} \...
So from what I've read, this theorem states that the total charge can be considered a point charge from outside the shell and that the field inside a uniform shell of charge is 0. Does this also apply to solid spheres?
Well I suppose I can write ##\varepsilon_{0} \vec{E} (4 \pi r^2) =...
I'm having trouble understanding how a charge distribution in a sphere can make this happen.
My instinct is that the fact that it's radially directed is a big hint of something, but I don't know what that hint might be alluding to. If the net E-field is constant inside the sphere and is always...
Also, was weight seen as the force due to gravity by Galileo's time? I've read that there was confusion among physicists at the time about the nature of weight; I wasn't sure if it was seen as synonymous with the gravitational force though.
Ahh so it was Newton who made the connection. But I still don't see how he'd figured a way to have a relative scale of forces. Did Newton use the weights of the bodies or something like spring scales to do this?
Also, based on DH's post Newton seemed to have related the change in momentum to...
But in order to make the proportion \frac{F_1}{m_1}\textit{=}\frac{F_2}{m_2}, as mikelepore stated, wouldn't force need to be quantified in some way? If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?
How did Galileo measure the forces? Did he equate the weight of the masses with the gravitational force?
For some reason I always assumed that, in Galileo's time, weight wasn't associated with the gravitational force; I just thought they had a very "primitive" understanding of what weight...
Sorry if this is considered necroposting! :rolleyes:
I just wanted to see if you could direct me to resources that show how Newton measured force and of Galileo and Newton's experiments that showed force being proportional to the product of mass and acceleration.
Yes, sorry, I should have included that as well.
Here's something I found though that explains the procedure a bit better (page 243, #389). What I don't understand though is why the two cases are taken consecutively (ie. the author goes from \frac{A}{a'} to \frac{a'}{a}). Is this the only way...
Unfortunately I have a very fragmented understanding of the very little I've read of the Elements. So basically, no I have not :frown: Forgive me!
I guess what I'm really asking is how concepts that involve mixed ratios (like velocity and density) came to be when their Euclidean roots forbade...
... Unless I'm mistaken and they did in fact represent physical quantities with numbers?
But if this was the case, why then did it take so long for mixed ratios like velocity and/or rate change to be accepted as viable comparisons? This is why I assumed that physical quantities were identified...
Ah okay, I see. It seems like the Ancient Greeks made it very clear in keeping magnitudes separate from numbers; they kept physical quantities separate from numbers and instead represented them with magnitudes (figures). Why did they do this though? What makes numbers so special that they felt...
Why aren't mixed ratios allowed in Euclid's "Elements"?
Why did Euclid forbid the comparing of different kinds of magnitudes? And was it the same for numbers? Or were ratios specifically meant for magnitudes (lines, planes, solids)
What are the consequences of comparing 2 different kinds of...