00:01
So for this problem, we are told that for the circular coil of radius art and entorns, carrying a current eye, the magnitude of the magnetic field at a point on its axis at a distance from its center is given by this expression right here.
00:20
So with that said, for part a of this problem, we are asked to show that this reduces to the familiar result for the field at the center.
00:32
Of the coil.
00:34
So in this case, we note that at the center of the coil, it states the value of zero.
00:40
So if we substitute that value into the previous expression, we reduce our expression of the magnetic field as mu -sub -0 times the current times the radius to the square times the number of thorns n.
00:53
And this divided by two times the radius square elevated to 3 divided by 2.
00:59
So now we can reduce this further, and now this simplifies with this two in here, and then we will obtain just simply the radius to the three.
01:11
Two times the radius to the three.
01:13
Now we can cancel this with this in here, and then finally we will obtain musup 0 times the current times number of thorns n, and this divided by two times the radius, which corresponds to the, familiar result for a field at the center of the coil.
01:35
Now, for part b of this problem, we are told to consider two parallel co -optial circular coils of equal radius capital r and number of thorns end, carrying equal currents in the same direction, and separated by a distance capital r.
01:55
Showed that the field on the axis around the midpoint between the coils is uniform over a distance that is small as compared to the radius capital art.
02:06
And that is given by the magnetic field is equal to 0 .72.
02:15
And this times mu -sub -0 times the number of thorns, times the current, and this divided by the radius.
02:32
So with that said, we note that in a small region, of length 2 times d.
02:40
The midpoint between the coils, we know that that midpoint is going to be a distance d from each coil.
02:47
So we can write that the magnetic field at that point is equal to mu -sub -0 times the current, times the radius square, times the number of thorns, and this divided by 2, and this times we can write the separation here.
03:05
Remember that this comes from the distance.
03:10
So that will be the radius divided by 2 plus the distance d, this to the square, and this plus the radius r to the square, all of this, elevated to the minus 3 divided by 2, and this plus the radios divided by 2 minus the distance d, all of this to the square, and this plus the radius capital r to the square, and this elevated to minus 3 divided by 2.
03:39
Now with that set, we can simplify some things in here.
03:46
So we can write that this is equal to musup 0 times the current times the radius to the 3, sorry, the radius squared times the number of terms this divided by 2.
03:59
And here we can take out a common term.
04:02
And that common term is going to be 5 times the radius squared divided by 4.
04:06
And this elevated to minus 3 divided by 2.
04:09
So once we have done that, we are left with the following.
04:23
That will be 1 plus 4 times d divided by 5 times the radios...