JEE Physics 4
Equal currents of magnitude 1 ampere are flowing through infinitely long wires parallel to y -axis located at x = 1 , 3 , 5 , … . The direction of the currents are alternative (the first one at x = 1 being positive) . Find the magnetic field at the origin .
If your answer can be represented in the form of μ 0 A where A is a positive real number, Find 8 A .
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1 solution
You may directly deduce even faster that the series n = 1 ∑ ∞ 2 n − 1 ( − 1 ) n − 1 = tan − 1 ( 1 )
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How did you get this ?
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Inspite of multiple methods , you could derive this as follows :
d x d ( tan − 1 x ) = 1 + x 2 1 = Infinite GP 1 − x 2 + x 4 − x 6 + ⋯
Next integrate both sides , tan − 1 x = x − 3 x 3 + ⋯ + C , Removing the constant by putting x = 0 we have , tan − 1 x = n = 1 ∑ ∞ 2 n − 1 ( − 1 ) n − 1 x n
Put x = 1 to obtain tan − 1 = 4 π
What if the current in the wires weren't alternative?
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@Kaushik Chandra – The Answer would diverge then
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@Aditya Narayan Sharma – That's true. But what will be the answer?
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@Kaushik Chandra – I said the 'answer' would diverge, that is the answer would be infinite
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@Aditya Narayan Sharma – Oh , I am sorry for the mistake . Sometimes, I don't know which language should I trust, English or Mathematics.
A nice amalgamation of pure mathematics and physics.