A red bulb and violet bulb of equal power emits `n_(R )` and `n_(v)` number of photons in a given time, then

Options

A. `n_R=n_V`
B. `n_R gt n_V`
C. `n_R lt n_V`
D. `n_R ge n_V`

Answer

To solve the problem of comparing the number of photons emitted by a red bulb and a violet bulb of equal power, we can follow these steps:

### Step 1: Understand the relationship between energy, power, and number of photons
The power of a light source is defined as the energy emitted per unit time. If both bulbs have equal power, we can express this relationship mathematically.

### Step 2: Write the formula for energy of a photon
The energy of a photon can be expressed using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength of the light.

### Step 3: Compare the wavelengths of red and violet light
We know that the wavelength of red light (\( \lambda_R \)) is greater than that of violet light (\( \lambda_V \)):
\[ \lambda_R > \lambda_V \]
This implies that:
\[ E_R < E_V \]
where \( E_R \) is the energy of a red photon and \( E_V \) is the energy of a violet photon.

### Step 4: Relate the number of photons to energy
Let \( n_R \) be the number of photons emitted by the red bulb and \( n_V \) be the number of photons emitted by the violet bulb. Since the power is the same for both bulbs, we can write:
\[ P = n_R \cdot E_R = n_V \cdot E_V \]

### Step 5: Rearrange the equation to find the ratio of the number of photons
From the equation above, we can express the ratio of the number of photons:
\[ \frac{n_R}{n_V} = \frac{E_V}{E_R} \]

### Step 6: Analyze the ratio
Since we established that \( E_V > E_R \), it follows that:
\[ \frac{E_V}{E_R} > 1 \]
This means:
\[ n_R > n_V \]

### Conclusion
Thus, the number of photons emitted by the red bulb (\( n_R \)) is greater than the number of photons emitted by the violet bulb (\( n_V \)). Therefore, the correct answer is:
\[ n_R > n_V \]

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