GATE 2026 EE Q.16: solution and answer

Official answer: BSource: Official answer key, GATE 2026 EE (IIT Guwahati)

1. Lorentz force on a charge $q>0$: $\mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times \mathbf{B}\right)$. The unit vectors ${\hat{a}}_x,{\hat{a}}_y,{\hat{a}}_z$ used below are the paper's $\hat{x},\hat{y},\hat{z}$, so $\mathbf{v}=5{\hat{a}}_x$, $\mathbf{E}=4{\hat{a}}_y$ and $\mathbf{B}=-6{\hat{a}}_z$.

2. Magnetic part: $\mathbf{v}\times \mathbf{B}=(5{\hat{a}}_x)\times (-6{\hat{a}}_z)=-30({\hat{a}}_x\times {\hat{a}}_z)$.

3. By the cyclic rule ${\hat{a}}_z\times {\hat{a}}_x={\hat{a}}_y$, so ${\hat{a}}_x\times {\hat{a}}_z=-{\hat{a}}_y$. This gives $\mathbf{v}\times \mathbf{B}=-30(-{\hat{a}}_y)=+30{\hat{a}}_y$.

4. Total force: $$\mathbf{F}=q\left(4{\hat{a}}_y+30{\hat{a}}_y\right)=34q{\hat{a}}_y$$

5. The force points along $+{\hat{a}}_y$, the same direction as $\mathbf{E}$. Since $\mathbf{B}$ is along ${\hat{a}}_z$, $\mathbf{F}\cdot \mathbf{B}=0$, so the force is perpendicular to the magnetic field.

6. Checking the other options: (D) is false, because the magnetic force $30q{\hat{a}}_y$ is not zero ($\mathbf{v}$ is perpendicular to $\mathbf{B}$). (A) and (C) are false, because the force has no ${\hat{a}}_z$ component and it is parallel to $\mathbf{E}$, not perpendicular to it. Answer: (B).