Question

An infinitely long wire is carrying a current $I$ in vertically upwards direction $(y-$direction$)$ as shown in figure.


At some time $t$, a particle of mass $m$ is at point $\text{A}$ and moving towards origin with velocity $v$. If the acceleration of the particle at point $\text{A}$ is $2g$ in vertically downward direction, then the nature of charge on the particle and its magnitude respectively are:

• A. $-ve;\frac{2\pi amg}{{\mu }_{0}vI}$
• B. $+ve;\frac{\pi amg}{{\mu }_{0}vI}$
• C. $-ve;\frac{\pi amg}{{\mu }_{0}vI}$
• D. $+ve;\frac{2\pi amg}{{\mu }_{0}vI}$

Answer 1

The correct option is D $+ve;\frac{2\pi amg}{{\mu }_{0}vI}$

Using right-hand thumb rule we obtained that direction of $\overrightarrow{B}$ at point $\text{A}$ due to infinite wire is along negative $z-$ direction $(⨂)$.

So, magnetic field produced by long wire at point $\text{A}$ will be

$\overrightarrow{B}=\frac{{\mu }_{0}I}{2\pi a}(-\hat{k})$

And velocity of the particles is $\overrightarrow{v}(-\hat{i})$.

The direction of force due to magnetic field on charge is,

$\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B}).....\left(1\right)$

Here $\overrightarrow{v}$ is along $(-\hat{i})$ and $\overrightarrow{B}$ is along $(-\hat{k})$.

Hence, the direction of $\overrightarrow{F}$ is along $(-\hat{i}\times -\hat{k})$ or $(-\hat{j})$, means vertically downward.


Also given, $\overrightarrow{a}=2g(-\hat{j})$

Therefore, from eq. $\left(1\right)$ the particle has a positive charge.

From FBD of particle and Newton's second law of motion,

${\overrightarrow{F}}_{net}=\overrightarrow{F}+{\overrightarrow{F}}_{grav}$

Since, $\overrightarrow{F}\&{\overrightarrow{F}}_{grav}$, both are in same direction

$\therefore |{\overrightarrow{F}}_{net}|=|\overrightarrow{F}|+|{\overrightarrow{F}}_{grav}|$

$m{a}_{net}=qvB\sin {90}^{\circ }+mg$

$m{a}_{net}=qvB+mg$

$2mg=qv\left(\frac{{\mu }_{0}I}{2\pi a}\right)+mg$

$\frac{{\mu }_{0}qvI}{2\pi a}=mg$

$\therefore q=\frac{2\pi amg}{{\mu }_{0}vI}$

Hence, option $\left(d\right)$ is the correct option.

Why this question ? It intends to test your understanding of the direction of $\overrightarrow{B},\overrightarrow{v}$ and $\overrightarrow{F_{mag}}$ on moving charge. Concept: The net acceleration of charged particle will be due to vector sum of all external forces acting on it. Tip: In this problem ${\overrightarrow{F}}_{net}$ on particle is its weight $\left(mg\right)$ and${\overrightarrow{F}}_{magnetic}$.