Two infinitely long wires are placed perpendicular to the plane of paper. Current in wire A to $$4$$ $$i0$$ outward the plane of paper and current in ‘B’ is $$i0$$ inward the plane of paper. The ∫− α$$+$$ αB→⋅ dℓ→ along the QP is $$K($$μ$$0i0)$$. Find the value of K.

Question

Two infinitely long wires are placed perpendicular to the plane of paper. Current in wire A to $$4$$ $$i0$$ outward the plane of paper and current in ‘B’ is $$i0$$ inward the plane of paper. The ∫− α$$+$$ αB→⋅ dℓ→ along the QP is $$K($$μ$$0i0)$$. Find the value of K.

Infinity Learn · Accepted Answer

Imagine a rectangular Amperian loop (cdefc) of very long length and very small breadth (almost zero).

$F o r f \to c : \int \vec{B} \cdot d \vec{ℓ} = x (s a y)$

$F o r c \to d : \int \vec{B} \cdot d \vec{ℓ} = 0 (M a g n e t i c f i e l d i s z e r o f o r l a r g e d i s \tan c e s)$

$F o r d \to e : \int \vec{B} \cdot d \vec{ℓ} = x (s y m m e t r y)$

$F o r e \to f : \int \vec{B} \cdot d \vec{ℓ} = 0 (M a g n e t i c f i e l d i s z e r o f o r l a r g e d i s \tan c e s)$

$∴ F o r l o o p c d e f c, u \sin g A m p e r e' s L a w, \oint \vec{B} \cdot d \vec{ℓ} = μ_{0} i_{e n c l o s e d}$

$\Rightarrow x + 0 + x + 0 = μ_{0} (4 i_{0})$

$\Rightarrow x = μ_{0} (2 i_{0})$

$\Rightarrow \int_{f}^{c} \vec{B} \cdot d \vec{ℓ} = 2 μ_{0} i_{0}$

Now, Imagine another rectangular Amperian loop (rstur) of very long length and very small breadth (almost zero).

$F o r r \to s : \int \vec{B} \cdot d \vec{ℓ} = x (s a y)$

$F o r s \to t : \int \vec{B} \cdot d \vec{ℓ} = 0 (M a g n e t i c f i e l d i s z e r o f o r l a r g e d i s \tan c e s)$

$F o r t \to u : \int \vec{B} \cdot d \vec{ℓ} = x (s y m m e t r y)$

$F o r u \to r : \int \vec{B} \cdot d \vec{ℓ} = 0 (M a g n e t i c f i e l d i s z e r o f o r l a r g e d i s \tan c e s)$

$∴ F o r l o o p r s t u r, u \sin g A m p e r e' s L a w, \oint \vec{B} \cdot d \vec{ℓ} = μ_{0} i_{e n c l o s e d}$

$\Rightarrow x + 0 + x + 0 = μ_{0} (i_{0}) \Rightarrow x = μ_{0} (0.5 i_{0})$

$\Rightarrow \int_{r}^{s} \vec{B} \cdot d \vec{ℓ} = 0.5 μ_{0} i_{0}$

$S i n c e, \int_{f}^{c} \vec{B} \cdot d \vec{ℓ} a n d \int_{r}^{s} \vec{B} \cdot d \vec{ℓ} a r e i n s a m e d i r e c t i o n, w e c a n a d d t h e m$

$∴ \int_{- α}^{+ α} \vec{B} \cdot d \vec{ℓ} = \int_{f}^{c} \vec{B} \cdot d \vec{ℓ} + \int_{r}^{s} \vec{B} \cdot d \vec{ℓ} = 2 μ_{0} i_{0} + 0.5 μ_{0} i_{0} = 2.5 μ_{0} i_{0}$

Magnetic field due to current element

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Magnetic field due to current element

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Two infinitely long wires are placed perpendicular to the plane of paper. Current in wire A to 4 i0 outward the plane of paper and current in ‘B’ is i0 inward the plane of paper. The ∫− α+ αB→⋅ dℓ→ along the QP is K(μ0i0). Find the value of K.

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