00:01
For this problem, we have an uniformly charged disk that has a radius capital art and surface charge density, as is shown in this figure.
00:11
So for part a of this problem, we are asked to find the electric potential point along the perpendicular central axis of the disk points.
00:24
So with that said, we need to find the amount of charge, the differential in charge, on a ring of radius.
00:32
And the width delta differential in the radius.
00:37
So we will have that that differentials in charge, as is shown here in the figure, is defined as the surface charge density times the differential in area.
00:52
So we can write this area as just simply 2 times pi times the radius times the differential in the radius.
01:00
So we will have that expression for the differential in the charge.
01:08
Now, to find the potential due to the ring, we will have the following expression.
01:17
So the potential is equal to columns constant, and this divided by the separation distance between this differential and the point p, which is this distance right here.
01:32
So that distance is going to be the square root of a square plus x square where a of course is some value in here and we need to integrate the differential in the charge oh sorry yes the radius oh we should have yes sorry the expression that we are going to use is this well we already know its form but the integral is going to be the integral of columns constant times the differential in charge this divided by the square root of the radios square plus the distance x square as is shown in here and then we can write the differential in charge as just simply well we'll have have close constant times 2 times pi times charge density times the radius, same to the differential in radius, and all of this divided by the square root of the radius square plus the distance x square.
03:05
So to obtain the total potential at the point pin, we need to integrate this expression over the limits from zero to the radius to the total radius.
03:19
Now what we can do is to take out everything that is common, that in everything that doesn't depend on the radius so we can take out pi times columns constant times the charge density then this the integral from zero to the total radius capital art these times two times the radius times the differential in the radios and this divided by the square root of the radius square plus the distance x square and then we know that in this case this simply integration will give us pi times columns comes sometimes sigma and in this case we will recognize that this will be the integral from zero to capital art of the radius square plus x squared that elevated to minus 1 divided by 2 this times two times the radius, this times the differential in the radius...
