JEE Main 2025 — 28 January, Morning Shift — Question 54

$\text{A}\to \text{B}$ $\vec{V}=-{{v}_{x}}\overset{}{\mathop{i}}\,+{{v}_{y}}\overset{}{\mathop{j}}\,$ $\text{vec{B}}=\frac{{{\mu }_{0}}\text{I}}{2\pi \text{r}}\left( -\overset{\text{}}{\mathop{\text{k}}}\, \right)$ \text{vec{F}}=\text{q}\left( \text{\vec{v}}\times \text{\vec{B}} \right)=\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{r}}\left[ -{{\text{v}}_{\text{x}}}\overset{\text{}}{\mathop{\text{j}}}\,-{{\text{v}}_{\text{y}}}\overset{\text{}}{\mathop{\text{i}}}\, \right] ${{\text{a}}_{\text{x}}}=-\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\cdot \frac{{{\text{v}}_{\text{y}}}}{\text{r}}$ ${{a}_{y}}=-\frac{{{\mu }_{0}}Iq}{2\pi m}\cdot \frac{{{v}_{x}}}{r}$ $\frac{{{\text{v}}_{\text{x}}}\text{d}{{\text{v}}_{\text{x}}}}{\text{dr}}=-\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\frac{{{\text{v}}_{\text{y}}}}{\text{r}}$ $\frac{{{\text{v}}_{\text{x}}}\text{d}{{\text{v}}_{\text{x}}}}{{{\text{v}}_{\text{y}}}}=-\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\frac{\text{dr}}{\text{r}}$ $\int _{0}^{{{v}_{0}}}\frac{{{v}_{x}}{{d}^{x}}{{v}_{x}}}{\sqrt{v_{0}^{2}-v_{x}^{2}}}=-\frac{{{\mu }_{0}}II{{q}^{{{x}_{1}}}}}{2\pi m}{{\int }_{a}}\frac{dr}{r}$ Let, ${{z}^{2}}={{v}_{0}}{{~}^{2}}-{{v}_{x}}{{~}^{2}}$ $2\text{zdz}=-2{{\text{v}}_{\text{x}}}\text{d}{{\text{v}}_{\text{x}}}$ $zdz=-{{v}_{x}}{{d}_{x}}$ $\frac{{{\text{v}}_{\text{x}}}\text{d}{{\text{v}}_{\text{x}}}}{\sqrt{\text{v}_{0}^{2}-\text{v}_{\text{x}}^{2}}}=\frac{-\text{zdz}}{\text{z}}=-\text{dz}$ then integral becomes $-\int _{{{\text{v}}_{0}}}^{0}\text{dz}=-\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\text{ln}\frac{{{\text{x}}_{1}}}{\text{a}}$ ${{\text{v}}_{0}}=-\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\text{ln}\frac{{{\text{x}}_{1}}}{\text{a}}$ ${{\text{X}}_{1}}=\text{a}{{\text{e}}^{-\frac{2\pi \text{m}{{\text{v}}_{0}}}{{{\mu }_{0}}\text{Iq}}}}$ For $\text{B}\to \text{C}$ $\vec{v}=-{{v}_{x}}\overset{}{\mathop{i}}\,-{{v}_{y}}\overset{}{\mathop{j}}\,$ \text{\vec{B}}=\frac{{{\mu }_{0}}\text{I}}{2\pi \text{r}}\left( -\overset{\text{}}{\mathop{\text{k}}}\, \right) $\vec{F}=q\left( \vec{v}\times \vec{B} \right)=\frac{{{\mu }_{0}}Iq}{2\pi r}\left( -{{v}_{x}}\overset{}{\mathop{j}}\,+{{v}_{y}}\overset{}{\mathop{i}}\, \right)$ ${{\text{a}}_{\text{x}}}=+\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\frac{{{\text{v}}_{\text{y}}}}{\text{r}}\text{ }\!\!~\!\!\text{ }{{\text{a}}_{\text{y}}}=-\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\cdot \frac{{{\text{v}}_{\text{x}}}}{\text{r}}$ $\frac{{{\text{v}}_{\text{x}}}\text{d}{{\text{v}}_{\text{x}}}}{\text{dr}}=\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\frac{{{\text{v}}_{\text{y}}}}{\text{r}}$ $\int _{{{v}_{0}}}^{0}\frac{{{v}_{x}}\text{d}{{\text{v}}_{\text{x}}}}{\sqrt{\text{v}_{0}^{2}-\text{v}_{\text{x}}^{2}}}=\frac{{{\mu }_{0}}\text{Iq}}{2\pi m}\int _{{{\text{x}}_{1}}}^{\text{x}}\frac{\text{dr}}{\text{r}}$ $\frac{{{\mu }_{0}}\text{Iq}}{2\pi \text{ }\!\!~\!\!\text{ m}}\text{ln}\frac{\text{x}}{{{\text{x}}_{1}}}=-\int _{0}^{{{\text{v}}_{0}}}\text{dz}=-{{\text{v}}_{0}}$ $\text{x}={{\text{x}}_{1}}{{\text{e}}^{-\frac{2\pi \text{m}{{\text{v}}_{0}}}{{{\mu }_{0}}\text{Iq}}}}\ldots$ From equation 1 and 2 $X=a{{e}^{-\frac{4\pi \text{m}{{\text{v}}_{0}}}{{{\mu }_{0}}{{\text{I}}_{\text{q}}}}}}$