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JEE Advance - Physics (2017 - Paper 1 Offline - No. 14)

A charged particle (electron or proton) is introduced at the origin (x=0,y=0,z=0) with a given initial velocity $$\overrightarrow v .$$ A uniform electric field $$\overrightarrow E $$ and a uniform magnetic field $$\overrightarrow B $$ exist everywhere. The velocity $$\overrightarrow v ,$$ electric field $$\overrightarrow E $$ and magnetic field $$\overrightarrow B $$ are given in column $$1,2$$ and $$3,$$ respectively. The quantities $${E_0},{B_0}$$ are positive in magnitude.

Column 1 Column 2 Column 3
(I) Electron with $$\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$$   (i) $$\overrightarrow E = {E_0}\widehat z$$ (P) $$\overrightarrow B = - {B_0}\widehat x$$
(II) Electron with $$\overrightarrow v = {{{E_0}} \over {{B_0}}}\widehat y$$ (ii) $$\overrightarrow E = - {E_0}\widehat y$$ (Q) $$\overrightarrow B = {B_0}\widehat x$$
(III) Proton with $$\overrightarrow v = 0$$    (iii) $$\overrightarrow E = - {E_0}\widehat x$$ (R) $$\overrightarrow B = {B_0}\widehat y$$
(IV) Proton with $$\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$$ (iv) $$\overrightarrow E = {E_0}\widehat x$$ (S) $$\overrightarrow B = {B_0}\widehat z$$
A charged particle (electron or proton) is introduced at the origin (x=0,y=0,z=0) with a given initial velocity $$\overrightarrow v .$$ A uniform electric field $$\overrightarrow E $$ and a uniform magnetic field $$\overrightarrow B $$ exist everywhere. The velocity $$\overrightarrow v ,$$ electric field $$\overrightarrow E $$ and magnetic field $$\overrightarrow B $$ are given in column $$1,2$$ and $$3,$$ respectively. The quantities $${E_0},{B_0}$$ are positive in magnitude.

Column 1 Column 2 Column 3
(I) Electron with $$\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$$   (i) $$\overrightarrow E = {E_0}\widehat z$$ (P) $$\overrightarrow B = - {B_0}\widehat x$$
(II) Electron with $$\overrightarrow v = {{{E_0}} \over {{B_0}}}\widehat y$$ (ii) $$\overrightarrow E = - {E_0}\widehat y$$ (Q) $$\overrightarrow B = {B_0}\widehat x$$
(III) Proton with $$\overrightarrow v = 0$$    (iii) $$\overrightarrow E = - {E_0}\widehat x$$ (R) $$\overrightarrow B = {B_0}\widehat y$$
(IV) Proton with $$\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$$ (iv) $$\overrightarrow E = {E_0}\widehat x$$ (S) $$\overrightarrow B = {B_0}\widehat z$$
A charged particle (electron or proton) is introduced at the origin (x=0,y=0,z=0) with a given initial velocity $$\overrightarrow v .$$ A uniform electric field $$\overrightarrow E $$ and a uniform magnetic field $$\overrightarrow B $$ exist everywhere. The velocity $$\overrightarrow v ,$$ electric field $$\overrightarrow E $$ and magnetic field $$\overrightarrow B $$ are given in column $$1,2$$ and $$3,$$ respectively. The quantities $${E_0},{B_0}$$ are positive in magnitude.

Column 1 Column 2 Column 3
(I) Electron with $$\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$$   (i) $$\overrightarrow E = {E_0}\widehat z$$ (P) $$\overrightarrow B = - {B_0}\widehat x$$
(II) Electron with $$\overrightarrow v = {{{E_0}} \over {{B_0}}}\widehat y$$ (ii) $$\overrightarrow E = - {E_0}\widehat y$$ (Q) $$\overrightarrow B = {B_0}\widehat x$$
(III) Proton with $$\overrightarrow v = 0$$    (iii) $$\overrightarrow E = - {E_0}\widehat x$$ (R) $$\overrightarrow B = {B_0}\widehat y$$
(IV) Proton with $$\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$$ (iv) $$\overrightarrow E = {E_0}\widehat x$$ (S) $$\overrightarrow B = {B_0}\widehat z$$
In which case would the particle move in a straight line along the negative direction of $$y$$-axis (i.e., move along $$ - \widehat y$$)?
$$\left( {{\rm I}{\rm I}} \right)\left( {iii} \right)\left( Q \right)$$
$$\left( {{\rm I}{\rm I}{\rm I}} \right)\left( {ii} \right)\left( R \right)$$
$$\left( {{\rm I}V} \right)\left( {ii} \right)\left( S \right)$$
$$\left( {{\rm I}{\rm I}{\rm I}} \right)\left( {ii} \right)\left( P \right)$$

Explanation

Lorentz force on a charged particle having a charge q, moving with a velocity $$\overrightarrow v $$ in a region of uniform electric field $$\overrightarrow E $$ and uniform magnetic field $$\overrightarrow B $$, is given by

$${\overrightarrow F _{net}} = {\overrightarrow F _E} + {\overrightarrow F _B} = q\overrightarrow E + q\overrightarrow v \times \overrightarrow B $$. ...... (1)

The particle will move along $$ - \widehat y$$ axis if (i) initial velocity $$\overrightarrow v $$ is along $$-\widehat y$$ and net force is either zero or directed along $$-\widehat y$$ (ii) initial velocity $$\overrightarrow v$$ is zero and net force is directed along $$-\widehat y$$. A proton with initial velocity $$\overrightarrow v = \overrightarrow 0 $$ placed in an electric field $$\overrightarrow E = - {E_0}\widehat y$$ and magnetic field $$\overrightarrow B = - {B_0}\widehat y$$ will experience a net force $${\overrightarrow F _{net}} = - q{E_0}\widehat y$$. It moves in a straight line along $$ - \widehat y$$.

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