A uniform magnetic field $\to B = (3^i + 4^j +^k)$ exists in region of space. A semicircular wire of radius $1 m$ carrying current $1 A$ 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 $\sqrt{2} (^i -^j +^k)$
R.E.F image
Length of $A B = 2 m$ (diameter of wire)
we white, reactor length of wire
$\to l = - - \to A B = 2 (cos 45^{0} \to i + sin 45^{0} \to j)$

$\Rightarrow \to l = \sqrt{2} (\to i + \to j)$
as magnetic field exists as $\to B = \to i + 4 \to j + \to k$ , force on wire would be:-
$\to F = I (\to i \times \to B)$ $[I = 1 A]$
$= 1 (\sqrt{2} (\to i + \to j)) \times (\to i + 4 \to j + \to k) = \sqrt{2} ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to i & \to j & \to k 1 & 1 & 0 1 & 4 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$\Rightarrow \to F = \sqrt{2} (\to i - \to j + 3 \to k)$

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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 1 A having its centre at (2, 2, 0) is placed in x−y plane as shown in figure. The force on semicircular wire will be

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