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Asked by minipkda 24th May 2018, 12:03 PM
Answered by Expert
Answer:
Let us consider a current carrying wire of length 2a through which a current of i is passing.
Let us consider a point P at a distance s from the wire.
Magnetic field dB due to a small current element idl is given by
begin mathsize 12px style stack d B with rightwards arrow on top space equals space fraction numerator mu subscript 0 over denominator 4 straight pi end fraction fraction numerator stack i d l with rightwards arrow on top cross times stack r subscript u with rightwards arrow on top over denominator r squared end fraction end style.......................(1)
where ru is the unit vector along r
from fig(a), we have l = s×tanθ , dl = s×sec2θ dθ ,  (s/r) = cosθ 
 
using the above relations in eqn. (1) , we have
 
begin mathsize 12px style d B space equals space fraction numerator mu subscript 0 i over denominator 4 pi s end fraction cos theta space d theta space end style..........................(2)
Magnetic field due to entire length of wire is obtained by integrating eqn.(2) as given below

begin mathsize 12px style B space equals fraction numerator mu subscript 0 i over denominator 4 pi s end fraction space integral subscript negative theta subscript 1 end subscript superscript theta subscript 2 end superscript cos theta space d theta space equals space fraction numerator mu subscript 0 i over denominator 4 pi s end fraction open parentheses sin theta subscript 2 space minus space sin left parenthesis negative theta subscript 1 right parenthesis close parentheses space equals space space fraction numerator mu subscript 0 i over denominator 4 pi s end fraction cross times fraction numerator 2 a over denominator square root of s squared plus a squared end root end fraction end style....................(3)
use this general relation (3) for the rectangular current loop shown in fig.(b) to get the magnetic field at centre of loop.
This magnetic field is given below
begin mathsize 12px style B space equals space fraction numerator 2 mu subscript 0 i over denominator pi a b end fraction square root of a squared plus b squared end root end style
 



Answered by Expert 25th May 2018, 10:41 PM
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