Question

In a certain mass spectrometer, the magnetic field has a magnitude of $0.2\text{T}$. It is intended that this spectrometer be used to separate two isotopes of Uranium, $U_{235}$ $($mass$=3.90\times {10}^{-25}\text{kg})$ and $U_{238}$ $($mass$=3.95\times {10}^{-25}\text{kg})$. In order to be separate the radii of curvature described by singly charged $($charge$+e)$ ions must differ by $2\text{mm}$. What will be the electric potential through which the ions must be accelerated in order to achieve this.

• A. $8\times {10}^4\text{V}$
• B. $4\times {10}^4\text{V}$
• C. $4\times {10}^2\text{V}$
• D. $2\times {10}^2\text{V}$

Answer 1

The correct option is A $8\times {10}^4\text{V}$

Let the isotopes $U_{235}$ and $U_{238}$ be denoted as $\left(1\right)$ and $\left(2\right)$ respectively.

Radius of the circular path $\left(r\right)$ is,

$r=\frac{mv}{Bq}=\frac{p}{Bq}$

$\therefore p=Bqr....\left(1\right)$

From $\left(1\right)$ we can write,

$p_1=Bq{r}_1$ and $p_2=Bq{r}_2$ and

$p_2-p_1=Bq{r}_2-Bq{r}_1=Bq(r_2-r_1)$

Substituting the values,

$p_2-p_1=\left(0.2\right)\times (1.6\times {10}^{-19})\times (2\times {10}^{-3})$

$p_2-p_1=6.4\times {10}^{-21}\text{kg}\text{m}/\text{s}........\left(2\right)$

From momentum-kinetic energy relation,

$K.E=\frac{p^2}{2m}$ also

$K.E=qV$

$\therefore qV=\frac{p^2}{2m}.........\left(3\right)$

From $\left(3\right)$ we can write,

$\therefore qV=\frac{p_1^2}{2m_1}=\frac{p_2^2}{2m_2}$

$\Rightarrow \frac{p_1^2}{m_1}=\frac{p_2^2}{m_2}$

$\Rightarrow p_2=p_1\sqrt{\frac{m_2}{m_1}}$

$\Rightarrow p_2=p_1\sqrt{\frac{3.95\times {10}^{-25}}{3.90\times {10}^{-25}}}$

$p_2=1.006p_1.....\left(4\right)$

Putting $\left(4\right)$ in $\left(2\right)$ we get,

$p_1={10}^{-19}\text{kg}\text{m}/\text{s}$

Now, from (3) we get,

$qV=\frac{p_1^2}{2m_1}$

$\Rightarrow V=\frac{p_1^2}{2m_{1}q}$

$\Rightarrow V=\frac{({10}^{-19}{)}^2}{2\times 3.90\times {10}^{-25}\times 1.6\times {10}^{-19}}$

$\therefore V=8\times {10}^4\text{V}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(a\right)$ is the correct answer.

Why this question ? This is a slightly complex example involving concepts from laws of motion and using magnetic field for particle separation.