Question

An X-ray tube operated at $50kV$, produces heat at the target at the rate of $796W$. If $0.5\% $ of the energy of electrons is used for X-ray generation, what is the number of electrons striking the target per second?
a. $ {10^{19}}$
b. ${10^{18}}$
c. ${10^{17}}$
d. ${10^{16}}$

Answer 1

Hint: Only some part of the total energy supplied is generated out as heat$(0.5\% )$ . The rest of the energy$(99.5\% )$ will be passed on to the electrons which will hit the target surface to produce the X-rays. Also, the stream of moving electrons can also be considered as current flowing.

Complete Step by Step Solution:
X-rays are produced due to sudden deceleration of fast-moving electrons when they collide and interact with the target anode. The major part of the energy is generated out as heat and the rest part is used for generating X-rays.

It is given that the heat generated$(H)$ is equal to $796W$ and $0.5\% $ of the total energy is generated out as heat.

Therefore, it can be rightly said that the remaining part of the total energy is generated as heat.
$H = 99.5\% $ of the total energy $ = 796W$ per second

Now, the total energy that is supplied per second is equal to the power supplied to the system.
Power supplied$(P) = VI$, where $V$ is the potential at which the X-ray tube is operated and $I$ is the current flowing.

It is given that $V = 50kV = 50 \times {10^3}V$
Since the current that is flowing actually consists of electrons as moving charges, the current$(I)$ can be expressed as a collection of several electrons. $I = ne$ , where $n$ is the number of electrons striking the target per second and $e$ is the charge of one electron $(e = 1.6 \times {10^{ - 19}}C)$.

 So, $P = V \times n \times e$

We can say that the $99.5\% $ of the total energy supplied per second is equal to $796W$.
$99.5\% $ of $P = 796W$
$\dfrac{{99.5}}{{100}} \times V \times n \times e = 796$
$\dfrac{{99.5}}{{100}} \times 50 \times {10^3} \times n \times 1.6 \times {10^{ - 19}} = 796$
$n = \dfrac{{796 \times 100}}{{99.5 \times 50 \times {{10}^3} \times 1.6 \times {{10}^{ - 19}}}}$
$n = 10 \times {10^{16}}$
$n = {10^{17}}$
So, Option $(C)$ is correct.

Note: The concepts used in the process of X-rays production are essential for problems like these. The student should carefully note that only some part of the total energy supplied is passed on to the electrons. Also, the relation between power and energy supplied may confuse the student, but a simple definition regarding this will help him/her to avoid any confusion.
The stream of moving electrons can be considered as current flowing and vice versa, so that the equations of power supplied can be applied in order to approach the problem.