A wire of arbitrary shape carries a current $ I = 2A $. Consider the portion of wire between $ (0, 0, 0) $ and $ (4, 4, 4) $. A magnetic field given by $ B = \left( 1.2 \times 10^{-4} + 2 \times 10^{-4} \right) \, \hat{k} $ exists in the region. The force acting on the given portion of the wire is:
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For a current-carrying wire in a magnetic field, the force is calculated using the cross product of the current direction and magnetic field.
The force on a current-carrying wire in a magnetic field is given by:
\[
F = I \cdot L \times B
\]
where \( I \) is the current, \( L \) is the length vector of the wire, and \( B \) is the magnetic field.
The magnetic field in the problem is:
\[
B = \left( 1.2 \times 10^{-4} + 2 \times 10^{-4} \right) \, \hat{k} = 3.2 \times 10^{-4} \, \hat{k}
\]
The length vector of the wire is from \( (0, 0, 0) \) to \( (4, 4, 4) \), so the length of the wire \( L = 4 \hat{i} + 4 \hat{j} + 4 \hat{k} \).
Using the cross-product for \( L \times B \) and solving for the force, we find the value of \( x \) in terms of time \( t \), giving the correct answer as \( x = 10.25 \sin(\omega t - \theta) \).
