00:01
All right, so let's say we have a sort of semicircular segment that is uniformly charged.
00:07
So let's say it has a total charge of plus q and it has a radius of capital r.
00:13
We want to know what's the voltage at the center.
00:16
So if we take a little infinitesimal slice q, the voltage is going to be k times dq over the distance from this point to the center, which is really just r, right? because for a semicircle, all the points are equidistant.
00:32
And so this is going to be basically k times q over r when we integrate over all the charges.
00:42
All right.
00:43
And then for the electric field, if we look at this, this will be the integral of k times dq.
00:50
And now the distance is still going to be r squared because they're all equidistant.
00:55
But we need to get the direction as well.
00:57
So we'll have an r cubed times r cosine theta ihat plus sine theta jhap.
01:08
And so what we can do is we'll integrate over our angles.
01:11
So we can write dq, it's just going to be r d theta.
01:15
And so we'll have k over r, our electric field, sorry, is k over r cubed.
01:22
And then we'll have a factor of r.
01:24
And then we're going from 0 to pi of the cosine of theta i -hat plus the sign of theta j -hat d -theta and so if you plug in these numbers this is k over r squared and then the integral of cosine is going to be a sine theta from zero to pi and then this is in the i -hat direction and then we have our integral here of minus the cosine of theta from 0 to pi j hat and so this will be k over r squared uh or sorry dq i i really apologize it's um it's q over r um or q over pi r that's the circumference times r d theta that's what i meant to write.
02:28
So it's really q over pi.
02:30
So i'm missing some extra terms...