In the below product, What will be the complete expression for B ?

The correct option is

Step-by-step Explanation:

From the image provided in the problem statement, we have the magnetic force equation:

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

where the magnetic field is given as:

$\overrightarrow{B}=B\hat{i}+B\hat{j}+B_0\hat{k}$

The given values are:

$q=1$

$\overrightarrow{v}=2\hat{i}+4\hat{j}+6\hat{k}$

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

Now, let us calculate the cross product $\overrightarrow{v}\times \overrightarrow{B}$:

$\overrightarrow{v}\times \overrightarrow{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}& 2& 4& 6& B& B& B_0\end{array}\right|$

Expanding the determinant:

$\overrightarrow{v}\times \overrightarrow{B}=\hat{i}(4B_0-6B)-\hat{j}(2B_0-6B)+\hat{k}(2B-4B)$

$\overrightarrow{v}\times \overrightarrow{B}=(4B_0-6B)\hat{i}+(6B-2B_0)\hat{j}-2B\hat{k}$

Since $q=1$, we have $\overrightarrow{F}=\overrightarrow{v}\times \overrightarrow{B}$. Equating the components of $\overrightarrow{F}$ with the calculated cross product:

1. From the $\hat{k}$ component:

$-2B=12\Rightarrow B=-6$

2. From the $\hat{i}$ component:

$4B_0-6B=4$

Substitute $B=-6$ into the equation:

$4B_0-6(-6)=4$

$4B_0+36=4\Rightarrow 4B_0=-32\Rightarrow B_0=-8$

3. Let us double-check using the $\hat{j}$ component:

$6B-2B_0=6(-6)-2(-8)=-36+16=-20$

This perfectly matches the $\hat{j}$ component of $\overrightarrow{F}$.

Substitute the values of $B=-6$ and $B_0=-8$ back into the magnetic field vector expression:

$\overrightarrow{B}=-6\hat{i}-6\hat{j}-8\hat{k}$

Thus, the complete expression for $\overrightarrow{B}$ is