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Force on a Finite Wire

A uniform mag...

Question

A uniform magnetic field $¯ B = (3^i + 4^j +^k)$ exists in region of space. A semicircular wire of radius 1 m carrying current 1A having its centre at (2, 2, 0) is placed in x-y plane as shown in figure. The force on semicircular wire will be

$\sqrt{2} (^i +^j +^k)$

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$\sqrt{2} (^i -^j +^k)$

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$\sqrt{2} (^i +^j -^k)$

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$\sqrt{2} (-^i +^j +^k)$

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Solution

The correct option is B
$\sqrt{2} (^i -^j +^k)$

Force on semicircular wire = force on section PQ.
$∴ \to F = i (\to l \times \to B)$
Here $\to l = - - \to P Q = (2 c o s 45^{\circ})^i + (2 s i n 45^{\circ})^j$
$= (\sqrt{2}^i + \sqrt{2}^j) m ∴ \to F = (1) [(\sqrt{2 i} + \sqrt{2 j})] \times (3^i + 4^j +^k) = \sqrt{2} (^i -^j +^k)$

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A uniform magnetic field ¯B=(3^i+4^j+^k) exists in region of space. A semicircular wire of radius 1 m carrying current 1A having its centre at (2, 2, 0) is placed in x-y plane as shown in figure. The force on semicircular wire will be

The correct option is B √2(^i−^j+^k) Force on semicircular wire = force on section PQ. ∴→F=i(→l×→B) Here →l=−−→PQ=(2 cos 45∘)^i+(2 sin 45∘)^j =(√2^i+√2^j)m∴→F=(1)[(√2i+√2j)]×(3^i+4^j+^k)=√2(^i−^j+^k)

A uniform magnetic field $¯ B = (3^i + 4^j +^k)$ exists in region of space. A semicircular wire of radius 1 m carrying current 1A having its centre at (2, 2, 0) is placed in x-y plane as shown in figure. The force on semicircular wire will be