AIIMS 2006 Physics X-rays MCQ Question

To determine the accelerating voltage required for producing hard X-rays with a minimum wavelength of $10^{-11}$ m, we can utilize the relationship between the energy of the X-rays and the wavelength, as well as the formula relating the accelerating voltage to the energy of the electrons.

Step 1: Understanding the Relationship Between Wavelength and Energy

The energy $E$ of a photon is related to its wavelength $\lambda$ by the equation:

$$E = \frac{hc}{\lambda}$$

where:

• $E$ is the energy of the photon,
• $h$ is Planck's constant ($6.626 \times 10^{-34} \, \text{Js}$),
• $c$ is the speed of light ($3 \times 10^8 \, \text{m/s}$),
• $\lambda$ is the wavelength.

Step 2: Calculating the Energy for the Minimum Wavelength

Substituting the values into the formula:

1. Given $\lambda = 10^{-11} \, \text{m}$:

$$E = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{10^{-11} \, \text{m}}$$

Calculating the energy:

$$E = \frac{1.9878 \times 10^{-25} \, \text{Js}}{10^{-11} \, \text{m}} = 1.9878 \times 10^{-14} \, \text{J}$$

Step 3: Converting Energy to Electron Volts

To relate this energy to an accelerating voltage $V$, we use the formula:

$$E = eV$$

where $e$ is the charge of an electron ($1.602 \times 10^{-19} \, \text{C}$). Rearranging gives:

$$V = \frac{E}{e}$$

Now substituting in the energy calculated above:

$$V = \frac{1.9878 \times 10^{-14} \, \text{J}}{1.602 \times 10^{-19} \, \text{C}} \approx 124.2 \, \text{kV}$$

Conclusion

From the calculation, we see that the minimum accelerating voltage required to produce X-rays with a wavelength of $10^{-11} \, \text{m}$ is approximately $124.2 \, \text{kV}$. Therefore, the accelerating voltage must be greater than $124.2 \, \text{kV}$ to ensure that X-rays of this wavelength can be produced, leading us to the correct answer, which is:

B) $> 124.2 \, \text{kV}$.

Explanation of Other Options

• A) $\leq 124.2 \, \text{kV}$: This option is incorrect because it suggests that a voltage less than or equal to $124.2 \, \text{kV}$ is sufficient, which we determined is not the case as it would not provide enough energy to produce X-rays of the required wavelength.

• C) between 60 kV and 70 kV: This is incorrect since the required voltage to produce the desired wavelength is much higher than this range.

• D) $> 100 \, \text{kV}$: While this option suggests a voltage that is indeed higher than $100 \, \text{kV}$, it does not guarantee that the voltage is sufficient to produce X-rays of the required wavelength, as it does not exceed $124.2 \, \text{kV}$.

In summary, the only correct and precise choice is B) $> 124.2 \, \text{kV}$, as calculated.