what is biot - savant law ? Derive an expression for magnetic field at a point on the axis of a current carrying circular coil
Answers
Conducting Coil
dB =
μ0
4π
idlsinφ
(a2 + x2)
As the loop lies perpendicular to the plane of paper and vector r→ in the plane of paper.
Hence angle φ between dl→ and r→ is 90o
∴ dB =
μ0idl
4π(a2 + x2)
Magnetic field dB→ can be resolved into two components one dBsinθ parallel to the axis of the loop and other dBcosθ perpendicular to the axis.
From the symmetry of the system it can be seen that diametrically opposite elements contribute to cancel the perpendicular components whereas parallel components are added up.
B = ∫ dBsinθ
Thus,
B = ∮
μ0
4π
idl
r2
sinθ
From the diagram we can observe:
r = √(a2 + x2 and sinθ = a/√(a2 + x2
∴ dB = ∮
μ0
4π
idl
(a2 + x2)
a
(a2 + x2)1/2
B =
μ0iα
4π(a2 + x2)3/2
∮ dl
B =
μ0iα
4π(a2 + x2)3/2
2πα
As we know area of circular coil is
A = πa2
∴ B =
μ0
4π
2iA
(a2 + x2)3/2
For coil with N turns -
∴ B =
μ0
4π
2NiA
(a2 + x2)3/2
We have Magnetic dipole moment of coil
=
∴ B =
μ0
4π
2M
(a2 + x2)3/2
Answer:
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Derived
and expression biont savart law and it's application