A charged particle (electron or proton) is introduced at the origin $(x=0, y=0, z=0)$ with a given initial velocity $\vec{v}$. A uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ exist everywhere. The velocity $\vec{v}$, electric field $\vec{E}$ and magnetic field $\vec{B}$ are given in columns 1, 2 and 3, respectively. The quantities $E_{0}, B_{0}$ are positive in magnitude. | Column 1 | Column 2 | Column 3 | | --- | --- | --- | | (I) Electron with $\vec{v}=2 \frac{E_{0}}{B_{0}} \widehat{x}$ | (i) $\vec{E}=E_{0} \hat{z}$ | (P) $\vec{B}=-B_{0} \hat{x}$ | | (II) Electron with $\vec{v}=\frac{E_{0}}{B_{0}} \hat{y}$ | (ii) $\vec{E}=-E_{0} \hat{y}$ | (Q) $\vec{B}=B_{0} \hat{x}$ | | (III) Proton with $\vec{v}=0$ | (iii) $\vec{E}=-E_{0} \hat{x}$ | (R) $\vec{B}=B_{0} \hat{y}$ | | (IV) Proton with $\vec{v}=2 \frac{E_{0}}{B_{0}} \widehat{x}$ | (iv) $\vec{E}=E_{0} \hat{x}$ | (S) $\vec{B}=B_{0} \hat{z}$ | In which case would the particle move in a straight line along the negative direction of $y$ axis (i.e., move along $-\hat{y}$ )?

JEE Advanced · 2017 · 21 May 2017 · Paper 1 · Magnetic Effects of Current

A charged particle (electron or proton) is introduced at the origin

(x = 0, y = 0, z = 0)

with a given initial velocity

v

. A uniform electric field

E

and a uniform magnetic field

B

exist everywhere. The velocity

v

, electric field

E

and magnetic field

B

are given in columns 1, 2 and 3, respectively. The quantities

E_{0}, B_{0}

are positive in magnitude.

Column 1	Column 2	Column 3
(I) Electron with $v = 2 \frac{E _{0}}{B _{0}} x$	(i) $E = E_{0} \overset{z}{^}$	(P) $B = - B_{0} \overset{x}{^}$
(II) Electron with $v = \frac{E _{0}}{B _{0}} \overset{y}{^}$	(ii) $E = - E_{0} \overset{y}{^}$	(Q) $B = B_{0} \overset{x}{^}$
(III) Proton with $v = 0$	(iii) $E = - E_{0} \overset{x}{^}$	(R) $B = B_{0} \overset{y}{^}$
(IV) Proton with $v = 2 \frac{E _{0}}{B _{0}} x$	(iv) $E = E_{0} \overset{x}{^}$	(S) $B = B_{0} \overset{z}{^}$

In which case would the particle move in a straight line along the negative direction of

y

axis (i.e., move along

- \overset{y}{^}

Answer: (B)

For motion along

- \overset{y}{^}

, the net force must be along

- \overset{y}{^}

and the particle must start from rest or have initial velocity along

y

. Option (B): (III) (ii) (R) corresponds to a proton with

v = 0

E = - E_{0} \overset{y}{^}

, and

B = B_{0} \overset{y}{^}

. The electric force on the proton is

F_{E} = e E = - e E_{0} \overset{y}{^}

, which accelerates it along

- \overset{y}{^}

. Since the velocity and magnetic field are parallel (both along

y

), the magnetic force is zero. Thus, the proton moves in a straight line along

- \overset{y}{^}

JEE Advanced · 2017 · 21 May 2017 · Paper 1 · Single correct option