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A charged particle carrying charge $1\,\mu C$ is moving with velocity $(2 \hat{ i }+3 \hat{ j }+4 \hat{ k })\, ms ^{-1} .$ If an external magnetic field of $(5 \hat{ i }+3 \hat{ j }-6 \hat{ k }) \times 10^{-3}\, T$ exists in the region where the particle is moving then the force on the particle is $\overline{ F } \times 10^{-9} N$. The vector $\overrightarrow{ F }$ is :
$-0.30 \hat{ i }+0.32 \hat{ j }-0.09 \hat{ k }$
$-300 \hat{ i }+320 \hat{ j }-90 \hat{ k }$
$-30 \hat{ i }+32 \hat{ j }-9 \hat{ k }$
$-3.0 \hat{ i }+3.2 \hat{ j }-0.9 \hat{ k }$
Solution
$\overrightarrow{ F }=q(\overrightarrow{ V } \times \overrightarrow{ B })$ (Force on charge particle moving in magnetic field)
$\overrightarrow{ V } \times \overrightarrow{ B }=(2 \hat{ i }+3 \hat{ j }+4 \hat{ k }) \times(5 \hat{ i }+3 \hat{ j }-6 \hat{ k }) \times 10^{-3}$
$=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 5 & 3 & -6\end{array}\right| \times 10^{-3}$
$=[\hat{i}[-18-12]-\hat{j}[-12-20]+\hat{k}[6-15]] \times 10^{-3}$
$=[\hat{i}[-30]+\hat{j}[32]+\hat{k}[-9]] \times 10^{-3}$
Force $=10^{-6}[-30 \hat{i}+32 \hat{j}-9 \hat{k}] \times 10^{-3}$
$=10^{-9}[-30 \hat{ i }+32 \hat{ j }-9 \hat{ k }]$
