Question

A positron with kinetic energy$\mathbf{2}\mathbf{.00}$keV is projected into a uniform magnetic field $\overrightarrow{\mathbf{B}}$of magnitude $\mathbf{0}\mathbf{.100}$T, with its velocity vector making an angle of 89.0° with.

(a) Find the period.

(b) Find the pitch p.

(c) Find the radius rof its helical path.

Short answer

a) The time period for positron will be .$3.58\times {10}^{-10}s$

b) The Pitch of the positron will be .$1.66\times {10}^{-4}m$

c) The radius of the helical path will be .$1.51\times {10}^{-3}m$

Step-by-step solution

Listing the given quantities

• $B=0.100T$
  

• The angle between velocity and the magnetic field is$\varphi =89.0°$
• $K=2.00keV$
  

Understanding the concept of magnetic force  

We are given the angle between the velocity of the positron and its magnetic field. From this, we can find the velocity component perpendicular to the magnetic field. We have the formula for the radius of the orbital of the positron. We substitute the radius and velocity component in the formula for the time period, and we get the time period. We are also given the kinetic energy of the positron; from this, we can find the velocity ofthepositron. Once we get the velocity, we can find the pitch and the radius of its helical path.

Formula:

$T=\frac{2\pi r}{V}$

$p=v\cos \varphi \times T$

$r=\frac{m_{e}v\sin \varphi }{eB}$

$K=\frac{1}{2}{m}_e{v}^2$

(a) Calculation oftime period for positron 

If$v$is the speed of the positron, then the speed of the positron along the plane perpendicular to the magnetic field will be, as the angle between velocity and field is given by.$\varphi$

We know that:

$T=\frac{2\pi r}{v}$

We substitute the value for r andin the above equation:

$\begin{array}{c}T=\frac{2\pi \left(\frac{m_{e}v\sin \varphi }{eB}\right)}{v\sin \varphi }\\ =\frac{2\pi m_e}{eB}\\ =\frac{2\pi (9.11\times {10}^{-31})}{(1.60\times {10}^{-19})(0.100)}\\ =3.58\times {10}^{-10}\text{\hspace{0.17em}s}\end{array}$

The time period for positron will be .$3.58\times {10}^{-10}s$

(b) Calculation ofpitch of the positron

As we know, the kinetic energy for the positron:

$K=\frac{1}{2}{m}_e{v}^2$

$v^2=\frac{2K}{m_e}$

$v=\sqrt{\frac{2K}{m_e}}$

We have to convert kinetic energy into Joule as:

$\begin{array}{c}K=2.00\times {10}^{3}eV\\ =2.00\times {10}^{3}eV\times \frac{1.60\times {10}^{-19}J}{1eV}\\ =3.2\times {10}^{-16}J\end{array}$

$\begin{array}{l}v=\sqrt{\frac{2\times 3.2\times {10}^{-16}}{9.11\times {10}^{-31}}}\\ v=2.65\times {10}^7\text{\hspace{0.17em}m/s}\end{array}$

We know that:

$\begin{array}{c}p=v\cos \varphi \times T\\ =2.65\times {10}^7\times cos89°\times 3.58\times {10}^{-10}\\ =1.66\times {10}^{-4}m\end{array}$

The Pitch of the positron will be .$1.66\times {10}^{-4}m$

(c) Calculation ofradius of the helical path

We know that:

$\begin{array}{c}r=\frac{m_{e}v\sin \varphi }{eB}\\ =\frac{9.11\times {10}^{-31}\times 2.65\times {10}^7\times sin89°}{1.60\times {10}^{-19}\times 0.100}\\ =1.51\times {10}^{-3}m\end{array}$

The radius of the helical pathwill be.$1.51\times {10}^{-3}m$