Q 12-04-235JEE MainJEE Main 2017 (8 Apr)Medium
In a certain region static electric and magnetic fields exist. The magnetic field is given by $\vec B = B_0(\hat i + 2\hat j - 4\hat k)$. If a test charge moving with a velocity $\vec v = v_0(3\hat i - \hat j + 2\hat k)$ experiences no force in that region, then the electric field in the region, in SI units, is:
Answer: (D) $\vec E = -v_0B_0(14\hat j + 7\hat k)$
No force means $q(\vec E + \vec v\times\vec B) = 0$, so $\vec E = -\vec v\times\vec B$.
$$\vec v\times\vec B = v_0B_0\begin{vmatrix}\hat i & \hat j & \hat k\\ 3 & -1 & 2\\ 1 & 2 & -4\end{vmatrix} = v_0B_0\left[(4 - 4)\hat i - (-12 - 2)\hat j + (6 + 1)\hat k\right] = v_0B_0(14\hat j + 7\hat k)$$
$$\vec E = -v_0B_0(14\hat j + 7\hat k)$$
Solution by Sreeraj P, M.Sc Physics