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Q85E

Expert-verifiedFound in: Page 67

Book edition
2nd Edition

Author(s)
Randy Harris

Pages
633 pages

ISBN
9780805303087

**An electron accelerated from rest through a potential difference V acquires a speed of 0.9998c. Find the value of V.**

The potential energy required to acquire a speed of 0.9998*c* is **25 Mega Volts.**

**The relativistic kinetic energy of a charged particle is equal magnitude to the change in potential energy as it is accelerated through a potential difference. **

$\begin{array}{c}KE\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Delta U\\ KE\text{\hspace{0.17em}}=\text{\hspace{0.17em}}q\Delta V\\ (\gamma \text{\hspace{0.17em}}-\text{\hspace{0.17em}}1){m}_{e}{c}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}e\Delta V\end{array}$ ...........(1)

Here,

- Lorentz factor $\begin{array}{rcl}\gamma & =& \frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}\\ & =& \frac{1}{\sqrt{1-\frac{{\left(0.9998c\right)}^{2}}{{c}^{2}}}}\\ & =& 50\\ & & \end{array}$
- Mass of electron ${m}_{e}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}9.1\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{-31}\text{\hspace{0.17em}}kg$
- Charge of electron $e\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1.6\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{-19}\text{\hspace{0.17em}}C$

Putting these values in above equation we get,

localid="1659084824833" $\begin{array}{c}(50\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1)\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}9.1\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{-31}\text{\hspace{0.17em}}kg\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{(3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{8}\text{\hspace{0.17em}}\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.)}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}(1.6\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{-19})\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\Delta V\\ \Delta V\text{\hspace{0.17em}}=\text{\hspace{0.17em}}25.08\text{\hspace{0.17em}}MV\end{array}$

So, to get an electron accelerated to a speed of 0.9998c, the required potential difference to be applied is 25 mega volts.

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