Two positive ions having the same charge q but different masses, m1 and m2 are accelerated horizontally from rest through a potential difference V. They then enter a region where there is uniform field B→ normal to the plane of the trajectory. If the beam entered the magnetic field along x-axis, the value of the y-coordinate for each ion at any time t is approximately ? provided x remain much smaller than y.

The path of ion is shown in fig.. The magnetic force provides an acceleration which is towards the center of path C. At any instant the accelerationac=Fm=qvBm , ay=accosθfor small x, θ→0,cosθ→1→ay≃ac=qvBmFor x = vt, ………………….(i) y=12ayt2 …………………..(ii)From above equations, we gety=12qBx2mv……………….(iii)Since ions are accelerated through potential V⇒12mv2=qV v=2qVm ⇒y=Bx2q8mV12

Two positive ions having the same charge q but different masses, m1 and m2 are accelerated horizontally from rest through a potential difference V. They then enter a region where there is uniform field B→ normal to the plane of the trajectory. If the beam entered the magnetic field along x-axis, the value of the y-coordinate for each ion at any time t is approximately ? provided x remain much smaller than y.

The path of ion is shown in fig.. The magnetic force provides an acceleration which is towards the center of path C. At any instant the accelerationac=Fm=qvBm , ay=accosθfor small x, θ→0,cosθ→1→ay≃ac=qvBmFor x = vt, ………………….(i) y=12ayt2 …………………..(ii)From above equations, we gety=12qBx2mv……………….(iii)Since ions are accelerated through potential V⇒12mv2=qV v=2qVm ⇒y=Bx2q8mV12

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Two positive ions having the same charge $q$ but different masses, $m_{1}$ and $m_{2}$ are accelerated horizontally from rest through a potential difference $V .$ They then enter a region where there is uniform field $\vec{B}$ normal to the plane of the trajectory. If the beam entered the magnetic field along $x -$ axis, the value of the $y$ -coordinate for each ion at any time $t$ is approximately ? provided $x$ remain much smaller than $y$ .

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$B x^{2} {(\frac{q}{8 m V})}^{\frac{1}{2}}$

$B x^{2} {(\frac{3 q}{8 m V})}^{\frac{1}{2}}$

$B x^{2} {(\frac{5 q}{8 m V})}^{\frac{1}{2}}$

$B x^{2} {(\frac{q}{4 m V})}^{\frac{1}{2}}$

answer is A.

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Detailed Solution

The path of ion is shown in fig.. The magnetic force provides an acceleration which is towards the center of path $C$ . At any instant the acceleration

$a_{c} = \frac{F}{m} = \frac{q v B}{m}, a_{y} = a_{c} \cos θ$