In the product For and and What will be the complete expression for ? [2021]

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Solution

Q.
In the product $\vec{F} = q (\vec{v} \times \vec{B}) = q \vec{v} \times (B \hat{i} + B \hat{j} + B_{0} \hat{k})$

For $q = 1$ and $\vec{v} = 2 \hat{i} + 4 \hat{j} + 6 \hat{k}$ and $\vec{F} = 4 \hat{i} - 20 \hat{j} + 12 \hat{k}$

What will be the complete expression for $\vec{B}$ [2021]

1 $6 \hat{i} + 6 \hat{j} - 8 \hat{k}$

2 $- 8 \hat{i} - 8 \hat{j} - 6 \hat{k}$

3 $- 6 \hat{i} - 6 \hat{j} - 8 \hat{k}$

4 $8 \hat{i} + 8 \hat{j} - 6 \hat{k}$

Ans.
(3)

Given: $\vec{v} = 2 \hat{i} + 4 \hat{j} + 6 \hat{k}$ and $q = 1; \vec{B} = B \hat{i} + B \hat{j} + B_{0} \hat{k}$

$\vec{F} = 4 \hat{i} - 20 \hat{j} + 12 \hat{k}$

Now, $\vec{v} \times \vec{B} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 4 & 6 \\ B & B & B_{0} \end{matrix} |$

$= \hat{i} (4 B_{0} - 6 B) - \hat{j} (2 B_{0} - 6 B) + \hat{k} (2 B - 4 B)$

Force   $\vec{F} = q (\vec{v} \times \vec{B})$

$4 \hat{i} - 20 \hat{j} + 12 \hat{k} = 1$ $[(4 B_{0} - 6 B) \hat{i} - (2 B_{0} - 6 B) \hat{j} - 2 B \hat{k}]$

By comparison:

   $4 B_{0} - 6 B = 4$ ...(i)

   $- 2 B = 12$

$\Rightarrow B = - 6$ ...(ii)

From eqn. (i) and (ii), $4 B_{0} - 6 (- 6) = 4$

$\Rightarrow 4 B_{0} + 36 = 4 \Rightarrow 4 B_{0} = - 32 \Rightarrow B_{0} = - 8$

So,   $\vec{B} = - 6 \hat{i} - 6 \hat{j} - 8 \hat{k}$

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