The effective mass of a photon with frequency `6.2 xx 10^15` Hz is

Options

A. `4.25 xx 10^(-35)` kg
B. `45.4 xx 10^(-36)` kg
C. `4.54 xx 10^(-35)` kg
D. `6.9 xx 10^(-20)` kg

Explanation

To find the effective mass of a photon with a frequency of \(6.2 \times 10^{15}\) Hz, we can use the relationship between energy and mass as described by Einstein's theory. Here’s a step-by-step solution:

### Step 1: Understand the relationship between energy and mass
According to Einstein's theory, the energy \(E\) of a photon can be expressed as:
\[ E = mc^2 \]
where \(m\) is the mass and \(c\) is the speed of light.

### Step 2: Use Planck's equation for energy
The energy of a photon can also be expressed using Planck's constant \(h\) and the frequency \(\nu\) of the photon:
\[ E = h \nu \]
where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)) and \(\nu\) is the frequency.

### Step 3: Set the two expressions for energy equal
Since both expressions represent the energy of the photon, we can set them equal to each other:
\[ mc^2 = h \nu \]

### Step 4: Solve for effective mass \(m\)
Rearranging the equation to solve for \(m\):
\[ m = \frac{h \nu}{c^2} \]

### Step 5: Substitute the known values
Now we can substitute the values of \(h\), \(\nu\), and \(c\) into the equation. The speed of light \(c\) is approximately \(3 \times 10^8 \, \text{m/s}\).

Substituting the values:
\[ m = \frac{(6.626 \times 10^{-34} \, \text{Js}) (6.2 \times 10^{15} \, \text{Hz})}{(3 \times 10^8 \, \text{m/s})^2} \]

### Step 6: Calculate the numerator
Calculating the numerator:
\[ 6.626 \times 10^{-34} \times 6.2 \times 10^{15} = 4.10612 \times 10^{-18} \, \text{J} \]

### Step 7: Calculate the denominator
Calculating the denominator:
\[ (3 \times 10^8)^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \]

### Step 8: Divide to find the mass
Now, substituting back into the equation for mass:
\[ m = \frac{4.10612 \times 10^{-18}}{9 \times 10^{16}} = 4.573 \times 10^{-35} \, \text{kg} \]

### Step 9: Express in scientific notation
This can be expressed as:
\[ m \approx 4.57 \times 10^{-35} \, \text{kg} \]

### Final Answer
The effective mass of the photon is approximately:
\[ m \approx 4.57 \times 10^{-35} \, \text{kg} \]

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