Answer:
Hi Carter,
The complete answer along with the explanation is shown below.
I hope it will clear your query
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Explanation:
The magnitude of the magnetic field on the axis of a circular loop, a distance z Ā from the loop center, is given by Eq.:
B = NĪ¼ĪæiRĀ² / 2(RĀ²+ZĀ²)Ā³Ć·Ā²
where
R is the radius of the loop
N is the number of turns
i is the current.
Both of the loops in the problem have the same radius, the same number of turns, Ā and carry the same current. The currents are in the same sense, and the fields they Ā produce are in the same direction in the region between them. We place the origin Ā at the center of the left-hand loop and let x be the coordinate of a point on the axis Ā between the loops. To calculate the field of the left-hand loop, we set z = x in the Ā equation above. The chosen point on the axis is a distance s ā x from the center of Ā the right-hand loop. To calculate the field it produces, we put z = s ā x in the Ā equation above. The total field at the point is therefore
B = NĪ¼ĪæiRĀ²/2 [1/ 2(RĀ²+xĀ²)Ā³Ć·Ā² Ā + 1/ 2(RĀ²+xĀ²-2sx+sĀ²)Ā³Ć·Ā²]
Its derivative with respect to x is
dB /dx= Ā - NĪ¼ĪæiRĀ²/2 [3x/ (RĀ²+xĀ²)āµĆ·Ā² Ā + 3(x-s)/(RĀ²+xĀ²-2sx+sĀ²)āµĆ·Ā² ]
When this is evaluated for x = s/2 (the midpoint between the loops) the result is
dB /dx= Ā - NĪ¼ĪæiRĀ²/2 [3(s/2)/ (RĀ²+sĀ²/4)āµĆ·Ā² Ā - 3(s/2)/(RĀ²+sĀ²/4)āµĆ·Ā² ] =0
independent of the value of s.
