A charged particle (electron or proton) is introduced at the origin $(x = 0, y = 0, x = 0)$ with a given initial velocity $\to v$ . A uniform electric field $\to E$ and a uniform magnetic field $\to B$ exist everywhere. The velocity $\to v$ , electric field $\to E$ and magnetic field $\to B$ 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 $\to v = 2 \frac{E_{0}}{B_{0}}^x$

$(i)$ $\to E = E_{0}^z$

$(P)$ $\to B = - B_{0}^x$

$(I I)$ Electron with $\to v = \frac{E_{0}}{B_{0}}^y$

$(i i)$ $\to E = - E_{0}^y$

$(Q)$ $\to B = - B_{0}^x$

$(I I I)$ Electron with $\to v = 0$

$(i i i)$ $\to E = - E_{0}^x$

$(R)$ $\to B = B_{0}^y$

$(I V)$ Electron with $\to v = 2 \frac{E_{0}}{B_{0}}^x$

$(i v)$ $\to E = E_{0}^x$

$(S)$ $\to B = B_{0}^z$

In which case will the particle move in a straight line with constant velocity?

(I V) (i) (S)

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(I I) (i i i) (S)

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(I I I) (i i i) (P)

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(I I I) (i i) (R)

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Solution

The correct option is B $(I I) (i i i) (S)$
For motion in a straght line it only possible in condition when, $\to V$ , $\to E$ and $\to B$ are mutually perpendicular
So, for this condition,
$q (\to V \times \to B) + q$
$\to E = 0$
$\to E = - \to V \times \to B$
or $\to E = - (\to V \times \to B)$

For option-A:
$E | | B$ but perpendicular to $\to V$
Hence path is helix with increasing pitch.

For option-B:
$\to V ⊥ \to E ⊥ \to B$ and $v = \frac{| E |}{| B |}$
Hence, it will travel in straight line so net flux on it is zero.
Hence, option $B$ is correct.

For C and D:
Force due to $\to B$ is zero. Hence, it will accelerate opposite to direction of $\to E$

So, from the above observation only option $B$ is correct.

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Similar questions

A charged particle (electron or proton) is introduced at the origin $(x = 0, y = 0, x = 0)$ with a given initial velocity $\to v$ . A uniform electric field $\to E$ and a uniform magnetic field $\to B$ exist everywhere. The velocity $\to v$ , electric field $\to E$ and magnetic field $\to B$ 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 $\to v = 2 \frac{E_{0}}{B_{0}}^x$	(i) $\to E = E_{0}^z$	(P) $\to B = - B_{0}^x$
(II) Electron with $\to v = \frac{E_{0}}{B_{0}}^y$	(ii) $\to E = - E_{0}^y$	(Q) $\to B = - B_{0}^x$
(III) Electron with $\to v = 0$	(iii) $\to E = - E_{0}^x$	(R) $\to B = B_{0}^y$
(IV) Electron with $\to v = 2 \frac{E_{0}}{B_{0}}^x$	(iv) $\to E = E_{0}^x$	(S) $\to B = B_{0}^z$

In which case will the particle describe a helical path with axis along the positive

z

direction?

Column 1	Column 2	Column 3
$(I)$ Electron with $\to v = 2 \frac{E_{0}}{B_{0}}^x$	$(i)$ $\to E = E_{0}^z$	$(P)$ $\to B = - B_{0}^x$
$(I I)$ Electron with $\to v = \frac{E_{0}}{B_{0}}^y$	$(i i)$ $\to E = - E_{0}^y$	$(Q)$ $\to B = - B_{0}^x$
$(I I I)$ Electron with $\to v = 0$	$(i i i)$ $\to E = - E_{0}^x$	$(R)$ $\to B = B_{0}^y$
$(I V)$ Electron with $\to v = 2 \frac{E_{0}}{B_{0}}^x$	$(i v)$ $\to E = E_{0}^x$	$(S)$ $\to B = B_{0}^z$

In which case will the particle move in a straight line with constant velocity?

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

In which case will the particle describe a helical path with axis along the positive

z

direction?

A charged particle is projected with velocity $\to v = u^i$ from the origin, in a region having electric field $\to E$ and magnetic field $\to B$ both. Value of $\to B$ and $\to E$ are given in column 1 and column 2 respectively and path of the particle is given in column 3.

$\begin{matrix} Column 1 & Column 2 & Column 3 (Magnetic field) & (Electric field) & (Path of particle) (i) & \to B = 0 & (i) & \to E = 0 & (P) & Circular path (i i) & \to B = B_{0}^i + B_{0}^j & (i i) & \to E = E_{0}^i & (Q) & Helical path (i i i) & \to B = B_{0}^j + B_{0}^k & (i i i) & \to E = B_{0} u (^j -^k) & (R) & Cycloid (i v) & \to B = B_{0}^k & (i v) & \to E = E_{0}^k & (S) & Straight line \end{matrix}$