The wavelength of a photon needed to remove a proton from a nucleus which is bound to the nucleus with 1MeV energy is nearly

Options

A. 1.2 nm
B. `1.2xx10^(-3)nm`
C. `1.2xx10^(-6)nm`
D. `1.2xx10^(1)nm`

Explanation

To find the wavelength of a photon needed to remove a proton from a nucleus that is bound with 1 MeV of energy, we can follow these steps:

### Step 1: Convert the Energy from MeV to Joules
The energy of the proton is given as 1 MeV. We need to convert this energy into joules for our calculations.

1 MeV = \(10^6\) eV
1 eV = \(1.6 \times 10^{-19}\) Joules

So,
\[ E = 1 \text{ MeV} = 10^6 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} \]
\[ E = 1.6 \times 10^{-13} \text{ J} \]

### Step 2: Use the Energy-Wavelength Relation
The energy of a photon can be related to its wavelength using the formula:
\[ E = \frac{hc}{\lambda} \]
Where:
- \(E\) is the energy of the photon (in joules)
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ J s}\))
- \(c\) is the speed of light (\(3 \times 10^8 \text{ m/s}\))
- \(\lambda\) is the wavelength (in meters)

### Step 3: Rearrange the Formula to Solve for Wavelength
Rearranging the formula to solve for wavelength (\(\lambda\)):
\[ \lambda = \frac{hc}{E} \]

### Step 4: Substitute the Values
Now substitute the known values into the equation:
\[ \lambda = \frac{(6.626 \times 10^{-34} \text{ J s})(3 \times 10^8 \text{ m/s})}{1.6 \times 10^{-13} \text{ J}} \]

### Step 5: Calculate the Wavelength
Now perform the calculation:
\[ \lambda = \frac{1.9878 \times 10^{-25} \text{ J m}}{1.6 \times 10^{-13} \text{ J}} \]
\[ \lambda \approx 1.241125 \times 10^{-12} \text{ m} \]
Converting meters to nanometers (1 m = \(10^9\) nm):
\[ \lambda \approx 1.241125 \times 10^{-12} \text{ m} \times 10^9 \text{ nm/m} \approx 1.241125 \times 10^{-3} \text{ nm} \]
Rounding this value gives:
\[ \lambda \approx 1.24 \times 10^{-3} \text{ nm} \]

### Final Answer
The wavelength of the photon needed to remove the proton is approximately \(1.24 \times 10^{-3} \text{ nm}\).

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