Question:
The masses of two particles having same kinetic energies are in the ratio \(2 : 1\). Then their de Broglie wavelengths are in the ratio.
The masses of two particles having same kinetic energies are in the ratio \(2 : 1\). Then their de Broglie wavelengths are in the ratio.
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Proportionalities for de Broglie wavelength \(\lambda\):
• For same velocity \(v\): \(\lambda \propto \frac{1}{m}\)
• For same kinetic energy \(K\): \(\lambda \propto \frac{1}{\sqrt{m}}\)
• For same accelerating potential \(V\): \(\lambda \propto \frac{1}{\sqrt{mq}}\)
• For same velocity \(v\): \(\lambda \propto \frac{1}{m}\)
• For same kinetic energy \(K\): \(\lambda \propto \frac{1}{\sqrt{m}}\)
• For same accelerating potential \(V\): \(\lambda \propto \frac{1}{\sqrt{mq}}\)
Updated On: May 13, 2026
- \(2:1\)
- \(1:2\)
- \(\sqrt{2}:1\)
- \(1:\sqrt{2}\)
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The Correct Option is D
Solution and Explanation
Concept:
The de Broglie wavelength \(\lambda\) of a particle is related to its momentum \(p\) by the equation \(\lambda = \frac{h}{p}\), where \(h\) is Planck's constant. Momentum can be expressed in terms of kinetic energy \(K\) and mass \(m\) as \(p = \sqrt{2mK}\).
Step 1: Establishing the relationship between wavelength, mass, and kinetic energy.
Substituting \(p = \sqrt{2mK}\) into the de Broglie equation gives: \[ \lambda = \frac{h}{\sqrt{2mK}} \]
Step 2: Formulating the ratio based on constant parameters.
The problem states that both particles have the same kinetic energy (\(K_1 = K_2 = K\)). Since \(h\) and \(K\) are constants, the wavelength is inversely proportional to the square root of the mass: \[ \lambda \propto \frac{1}{\sqrt{m}} \] Therefore, the ratio of their wavelengths is: \[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2}{m_1}} \]
Step 3: Substituting the given mass ratio.
The given mass ratio is \(m_1 : m_2 = 2 : 1\), which means: \[ \frac{m_1}{m_2} = \frac{2}{1} \implies \frac{m_2}{m_1} = \frac{1}{2} \] Substitute this into the wavelength ratio equation: \[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{1}{2}} \] \[ \frac{\lambda_1}{\lambda_2} = \frac{1}{\sqrt{2}} \]
Step 4: Selecting the correct option.
The ratio of their de Broglie wavelengths is: \[ \boxed{1:\sqrt{2}} \] Therefore, the correct option is: \[ \boxed{\text{Option (D)}} \]
Step 1: Establishing the relationship between wavelength, mass, and kinetic energy.
Substituting \(p = \sqrt{2mK}\) into the de Broglie equation gives: \[ \lambda = \frac{h}{\sqrt{2mK}} \]
Step 2: Formulating the ratio based on constant parameters.
The problem states that both particles have the same kinetic energy (\(K_1 = K_2 = K\)). Since \(h\) and \(K\) are constants, the wavelength is inversely proportional to the square root of the mass: \[ \lambda \propto \frac{1}{\sqrt{m}} \] Therefore, the ratio of their wavelengths is: \[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2}{m_1}} \]
Step 3: Substituting the given mass ratio.
The given mass ratio is \(m_1 : m_2 = 2 : 1\), which means: \[ \frac{m_1}{m_2} = \frac{2}{1} \implies \frac{m_2}{m_1} = \frac{1}{2} \] Substitute this into the wavelength ratio equation: \[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{1}{2}} \] \[ \frac{\lambda_1}{\lambda_2} = \frac{1}{\sqrt{2}} \]
Step 4: Selecting the correct option.
The ratio of their de Broglie wavelengths is: \[ \boxed{1:\sqrt{2}} \] Therefore, the correct option is: \[ \boxed{\text{Option (D)}} \]
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