What is the ratio of the de Broglie wavelength for electrons accelerated through 50 volts and 200 volts?

Options

A. `3 : 10`
B. `10 : 3`
C. `1:2`
D. `2 : 1`

Answer

To find the ratio of the de Broglie wavelength for electrons accelerated through 50 volts and 200 volts, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \(h\) is Planck's constant and \(p\) is the momentum of the electron.

### Step 2: Relate momentum to kinetic energy
The momentum \(p\) can also be expressed in terms of kinetic energy (K):
\[ p = \sqrt{2mK} \]
where \(m\) is the mass of the electron and \(K\) is its kinetic energy.

### Step 3: Express kinetic energy in terms of voltage
The kinetic energy of an electron accelerated through a potential difference \(V\) is given by:
\[ K = eV \]
where \(e\) is the charge of the electron. Therefore, we can substitute this into the momentum equation:
\[ p = \sqrt{2meV} \]

### Step 4: Substitute into the de Broglie wavelength formula
Substituting the expression for momentum into the de Broglie wavelength formula, we get:
\[ \lambda = \frac{h}{\sqrt{2meV}} \]

### Step 5: Find the ratio of wavelengths for different voltages
Now, we want to find the ratio of the de Broglie wavelengths for electrons accelerated through 50 volts (λ₁) and 200 volts (λ₂):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{2meV_2}}{\sqrt{2meV_1}} = \sqrt{\frac{V_2}{V_1}} \]
where \(V_1 = 50\) volts and \(V_2 = 200\) volts.

### Step 6: Calculate the ratio
Substituting the values:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{200}{50}} = \sqrt{4} = 2 \]
Thus, the ratio of the de Broglie wavelengths is:
\[ \lambda_1 : \lambda_2 = 2 : 1 \]

### Final Answer
The ratio of the de Broglie wavelength for electrons accelerated through 50 volts and 200 volts is \(2 : 1\).

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