Two infinitely long wires are placed perpendicular to the plane of paper. Current in wire A to $$4i0$$ 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 $$4i0$$ 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).

$\begin{array}{l} F o r f \to c : \int \vec{B} \cdot d \vec{l} = x (s a y) \\ F o r c \to d : \int \vec{B} \cdot d \vec{l} = 0 (M a g n e t i c f i e l d i s z e r o f o r l a r g e d i s \tan c e s) \\ F o r d \to e : \int \vec{B} \cdot d \vec{l} = x (s y m m e t r y) \\ F o r e \to f : \int \vec{B} \cdot d \vec{l} = 0 (M a g n e t i c f i e l d i s z e r o f o r l a r g e d i s \tan c e s) \\ ∴ F o r l o o p c d e f c, u \sin g A m p e r e' s L a w, \oint \vec{B} \cdot d \vec{l} = μ_{0} i_{e n c l o s e d} \\ \Rightarrow x + 0 + x + 0 = μ_{0} (4 i_{0}) \\ \Rightarrow x = μ_{0} (2 i_{0}) \\ \Rightarrow \int_{f}^{c} \vec{B} \cdot d \vec{l} = 2 μ_{0} i_{0} \end{array}$

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

$\begin{array}{l} F o r r \to s : \int \vec{B} \cdot d \vec{l} = x (s a y) \\ F o r s \to t : \int \vec{B} \cdot d \vec{l} = 0 (M a g n e t i c f i e l d i s z e r o f o r l a r g e d i s \tan c e s) \\ F o r t \to u : \int \vec{B} \cdot d \vec{l} = x (s y m m e t r y) \\ F o r u \to r : \int \vec{B} \cdot d \vec{l} = 0 (M a g n e t i c f i e l d i s z e r o f o r l a r g e d i s \tan c e s) \\ ∴ F o r l o o p r s t u r, u \sin g A m p e r e' s L a w, \\ \oint \vec{B} \cdot d \vec{l} = μ_{0} i_{e n c l o s e d} \\ \Rightarrow x + 0 + x + 0 = μ_{0} (i_{0}) \\ \Rightarrow x = μ_{0} (0.5 i_{0}) \\ \Rightarrow \int_{r}^{s} \vec{B} \cdot d \vec{l} = 0.5 μ_{0} i_{0} \end{array}$

$\begin{array}{l} S i n c e, \int_{f}^{c} \vec{B} \cdot d \vec{l} a n d \int_{r}^{s} \vec{B} \cdot d \vec{l} a r e i n s a m e d i r e c t i o n, w e c a n a d d t h e m \\ ∴ \int_{- α}^{+ α} \vec{B} \cdot d \vec{l} = \int_{f}^{c} \vec{B} \cdot d \vec{l} + \int_{r}^{s} \vec{B} \cdot d \vec{l} = 2 μ_{0} i_{0} + 0.5 μ_{0} i_{0} = 2.5 μ_{0} i_{0} \end{array}$

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 $i_{0}$ outward the plane of paper and current in ‘B’ is $i_{0}$ inward the plane of paper. The $\int_{- α}^{+ α} \vec{B} \cdot d \vec{l}$ along the QP is $K (μ_{0} i_{0})$ . Find the value of K.

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Magnetic field due to current element

Remember concepts with our Masterclasses.

Two infinitely long wires are placed perpendicular to the plane of paper. Current in wire A to 4i0 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.

Ready to Test Your Skills?

free study material

Two infinitely long wires are placed perpendicular to the plane of paper. Current in wire A to 4 $i_{0}$ outward the plane of paper and current in ‘B’ is $i_{0}$ inward the plane of paper. The $\int_{- α}^{+ α} \vec{B} \cdot d \vec{l}$ along the QP is $K (μ_{0} i_{0})$ . Find the value of K.