Magnetic field on the axis of a circular current loop derivation 2
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According to the Biot Savarts Law the magnitude of the magnetic field induction at point P due to the small elements are equal ,that is ,
dB =μ04πIdl sin90°r2=μ04πIdl R2+x2since the angle is small therefore the angle is of 90°.The total magnetic induction at P due to current through the whole circular oil is given byB =∫dBsinθ=∫μ04πIdl sinθR2+x2=μ0I4πR2+x2sinθ∫dlSince sinθ=RR2+x2 and ∫dl=2πR∴B =μ0I4πR2+x2RR2+x22πR=μ04π2πR2IR2+x232If there are n turns in the coil then ∫dl=2πRnSo, the magnetic field induction B=μ04π2πnR2IR2+x232
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dB =μ04πIdl sin90°r2=μ04πIdl R2+x2since the angle is small therefore the angle is of 90°.The total magnetic induction at P due to current through the whole circular oil is given byB =∫dBsinθ=∫μ04πIdl sinθR2+x2=μ0I4πR2+x2sinθ∫dlSince sinθ=RR2+x2 and ∫dl=2πR∴B =μ0I4πR2+x2RR2+x22πR=μ04π2πR2IR2+x232If there are n turns in the coil then ∫dl=2πRnSo, the magnetic field induction B=μ04π2πnR2IR2+x232
Regards
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