Question

An X-ray tube, operated at a potential difference of 40 kV, produce heat at the rate of 720 W. Assuming 0.5% of the energy of incident electrons is converted into X-rays, calculate
(i)The number of electrons per second striking the target (ii)The velocity of the incident electrons .

Answer 1

Hint: X- ray is a light but it has more power and energy than light detected by our eyes. If electrons will strike at X ray or any other ray then some amount of electron which the electron has is transferred to the X- ray and untransfered energy will remain as an electron. To calculate energy use a simple formula of kinetic energy which must be equal to electron volt. Use the formula of power.

Complete step by step answer:

Potential difference is given by 40kV operated on X-ray tubes.
And heat produced is 720W
0.5 % of the energy of the incident electron is converted into X- ray. Then the remaining energy the electron has is 99.5% i.e. 0.995. it means 99.5% remain from total power.
Mathematically, power is given by,
P = 0.995 VI
Therefore,
$I=\dfrac{P}{0.995\times V}$
Put values of power i.e 720W and potential difference (p) is 40Kv in above equations,
We get,
$\begin{align}
  & I=\dfrac{720}{0.995\times 40\times {{10}^{3}}} \\
 & I=0.018 \\
\end{align}$
(i) The number of electron per second striking the target
Number of electrons is given by,
$\begin{align}
  & Number\text{ }of\text{ }electrons\text{ }=\dfrac{current}{charge on electron} \\
 & n=\dfrac{I}{q} \\
 & n=\dfrac{0.018}{1.6\times {{10}^{-19}}} \\
 & n=1.1\times {{10}^{17}}electrons \\
\end{align}$
The number of electron per second striking the target $n=1.1\times {{10}^{17}}electrons$
(ii)The velocity of the incident electrons.
Energy of incident electron is given by,
Use formula of kinetic energy then,
\[\begin{align}
  & \dfrac{1}{2}m{{v}^{2}}=eV \\
 & V=\sqrt{\dfrac{2eV}{m}} \\
 & V=\sqrt{{{\dfrac{2\times 1.6\times 40\times {{10}^{3}}\times 10}{9.1\times {{10}^{-31}}}}^{-19}}}m/\sec \\
 & V=1.2\times {{10}^{8}}m/\sec \\
\end{align}\]
The velocity of the incident electrons is \[V=1.2\times {{10}^{8}}m/\sec \].

Note: First understand what the question wants to convey then try to analyse the solution. X-ray are electromagnetic waves of very short wavelength. Use a direct formula of energy. Number of electrons is equal to the current by charge of the electron. Use a formula of power which is equal to the product of voltage and current.