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Question:

Using Biot-Savart's law, derive the expression for the magnetic field in the vector form at a point on the axis of a circular current loop. What does a toroid consist of?

Solution:

I→current
R→Radii
X→Axis
x→Distance of OP
dl→Conducting element of the loop
According to Biot-Savart's law, the magnetic field at P is
dB=μ₀I|dl×r|/4πr³
r²=x²+R²
|dl×r|=rdl(they are perpendicular)
∴dB=μ₀/4πIdl/(x²+R²)
dB has two components −dBx and dB⊥. dB⊥ is cancelled out and only the x-component remains
∴dBx=dBcosθ
cosθ=R/(x²+R²)¹/²
dBx=μ₀IdlR/4π(x²+R²)³/²
Summation of dl over the loop is given by 2πR
∴B=Bx i=μ₀IR²/2(x²+R²)³/² i
(b) Toriod is a hollow circular ring on which a large number of turns of wire are closely wound.
Three circular Amperian loops 1,2 and 3 are shown by dashed lines.
By symmetry, the magnetic field should be tangential to each of them and constant in magnitude for a given loop.
Let the magnetic field inside the toroid be B. We shall now consider the magnetic field at S.
By Ampere's Law, ∮B⋅dl=μ₀I
BL=μ₀NI
Where L is the length of the loop for which B is tangential I be the current enclosed by the loop and N be the number of turns.
We find, L=2πr
The current enclosed I is NI
B(2πr)=μ₀NI, therefore, B=μ₀NI/2πr
For a loop inside the toroid, no current exists thus, I=0 Hences, B=0
Exterior to the toroid :
Each turn of current coming out of the plane of the paper is cancelled exactly by the current going into it. Thus I=0, and, B=0