Calculating Shortest X-ray Wavelength
The shortest wavelength ($\lambda_{min}$) of X-rays produced is determined by the maximum energy of the incident electrons, which corresponds to the accelerating potential applied. This relationship is described by the Duane-Hunt law.
Duane-Hunt Law Principle
The maximum energy gained by an electron accelerated through a potential $V$ is converted into a single photon of maximum frequency (minimum wavelength). The energy conservation equation is:
$ E_{electron} = E_{photon} $
$ eV = \frac{hc}{\lambda_{min}} $
Where:
- $e$ = elementary charge ($1.602 \times 10^{-19}$ C)
- $V$ = accelerating potential ($50 \text{ KV} = 50 \times 10^3 \text{ V}$)
- $h$ = Planck's constant ($6.626 \times 10^{-34} \text{ J}\cdot\text{s}$)
- $c$ = speed of light ($3 \times 10^8 \text{ m/s}$)
- $\lambda_{min}$ = shortest wavelength
Calculation Steps
- Rearrange the formula to solve for $\lambda_{min}$:
$ \lambda_{min} = \frac{hc}{eV} $
- Use the convenient form $hc \approx 12400 \text{ eV} \cdot \text{\AA}$:
$ \lambda_{min} = \frac{12400 \text{ eV} \cdot \text{\AA}}{V (\text{in Volts})} $
Note: When $V$ is in kilovolts (KV), the formula becomes $\lambda_{min} = \frac{12.4 \text{ keV} \cdot \text{\AA}}{V (\text{in kV})}$.
- Substitute the given accelerating potential ($V = 50$ KV):
$ \lambda_{min} = \frac{12.4 \text{ keV} \cdot \text{\AA}}{50 \text{ kV}} $
- Calculate the minimum wavelength:
$ \lambda_{min} = 0.248 \text{\AA} $
- Rounding the result gives $0.25 \text{\AA}$.
Conclusion
The shortest wavelength present in the X-rays is approximately $0.25 \text{\AA}$. This corresponds to Option C.
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Important Questions from Mixed Topic (CUET PG)
- Who was the founder of Bolshevik Communist party?
- What is the key guide to statecraft in the realist tradition?
- Chronologically arrange the events in the Cold War period.
A. Berlin Wall is constructed
B. Communist China joins the UN
C. Soviet invasion of Czechoslovakia
D. Berlin Blockade
Choose the correct answer from the options given below: Morgenthau's principles of political realism are:
A. Politics is rooted in permanent and unchanging human nature which is basically self centred, self-regarding and self-interested
B. Politics is an autonomous sphere of action and cannot therefore be reduced to morals
C. International Politics is an arena of conflicting self-interests
D. The ethics of international relations is situational ethics which is very different from private morality
Choose the correct answer from the options given below:Who among the following political thinkers consider the anarchical self help system to be a compelling factor for States to maximise their relative power positions?
