The Question is from neet-physics atoms-nuclei x-rays, and it was created by the author NAGA SAI

Question

An X-ray tube operates at 40 kV. The short wavelength limit of the continuous X-ray spectrum emitted is closest to:

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Duane-Hunt Law
	**Related Topics**: X-ray production, Electromagnetic spectrum, Energy quantization
	**Prerequisites**: Relationship between energy, wavelength, and frequency of photons; electron-volt unit.
## SOLUTION PATHWAY
**Given Information**:
	Operating voltage (V) = 40 kV = $40 	imes 10^3$ V
**Method & Steps**:
	1. **Duane-Hunt Law**: The minimum wavelength ($\lambda_{min}$) of X-rays produced is inversely proportional to the accelerating voltage (V).
		Reasoning: The maximum energy of an X-ray photon is equal to the kinetic energy of the incident electron, which is determined by the accelerating voltage.
		Formula/Rule: $\lambda_{min} = \frac{hc}{eV}$, where h is Planck's constant ($6.626 	imes 10^{-34}$ Js), c is the speed of light ($3 	imes 10^8$ m/s), e is the elementary charge ($1.602 	imes 10^{-19}$ C), and V is the voltage.
	2. **Calculation**:
		Working: $\lambda_{min} = \frac{(6.626 	imes 10^{-34} Js)(3 	imes 10^8 m/s)}{(1.602 	imes 10^{-19} C)(40 	imes 10^3 V)}$
		Key Point: Ensure consistent units (Joules, Coulombs, Volts, meters).
	3. **Result**:
		Result: $\lambda_{min} \approx 3.1 	imes 10^{-11}$ m = 0.31 Å
		Verification: The calculated minimum wavelength is consistent with the Duane-Hunt law, showing an inverse relationship with the voltage.
## OPTION ANALYSIS
[A] ✓: '0.31 Å': This is the correct answer, derived directly from the Duane-Hunt law using the given voltage.
[B] :x: '0.62 Å': This value is double the correct answer, likely resulting from an incorrect application or misunderstanding of the Duane-Hunt law.
[C] :x: '0.15 Å':  This value is half the correct answer, possibly due to a calculation error or misinterpretation of the relationship between wavelength and voltage.
[D] :x: '0.47 Å': This value does not correspond to any simple error and likely represents a misunderstanding of the underlying physics.
## LEARNING AIDS
	**Common Mistakes**:
		1. Incorrect unit conversion: Ensure proper conversion between electron-volts, Joules, and other units.
		2. Misapplication of the Duane-Hunt law: Remember the inverse relationship between minimum wavelength and voltage.
	**Quick Tips**:
		1. Use the simplified formula: $\lambda_{min} (Å) \approx \frac{12.4}{V (kV)}$ for quick estimation.
		2. Remember that higher voltage leads to shorter wavelengths in the X-ray spectrum.

NAGA SAI · Answer

The correct Option is "0.31 Å"

NAGA SAI · Answer

0.62 Å

NAGA SAI · Answer

0.15 Å

NAGA SAI · Answer

0.47 Å

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