A laser produces 20.0 mW of red light. In 30.0 minutes, the laser emits 1.15×10²⁰ photons. What is the wavelength of the laser?

검증된 단계별 안내

Convert the power of the laser from milliwatts to watts. Since 1 mW = 0.001 W, multiply 20.0 mW by 0.001 to get the power in watts.

Calculate the total energy emitted by the laser in 30.0 minutes. Use the formula: \( \text{Energy} = \text{Power} \times \text{Time} \). Convert 30.0 minutes to seconds by multiplying by 60.

Determine the energy of a single photon. Use the formula: \( E_{\text{photon}} = \frac{\text{Total Energy}}{\text{Number of Photons}} \).

Use the energy of a single photon to find the wavelength. Use the formula: \( E_{\text{photon}} = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J s} \) and \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \). Rearrange to solve for \( \lambda \): \( \lambda = \frac{hc}{E_{\text{photon}}} \).

Substitute the values for \( h \), \( c \), and \( E_{\text{photon}} \) into the equation to calculate the wavelength \( \lambda \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Photon Energy

The energy of a photon is given by the equation E = hν, where E is energy, h is Planck's constant (6.626 × 10^-34 J·s), and ν (nu) is the frequency of the light. This relationship shows that the energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength.

Wavelength and Frequency Relationship

The wavelength (λ) and frequency (ν) of light are related by the equation c = λν, where c is the speed of light (approximately 3.00 × 10^8 m/s). This means that as the wavelength increases, the frequency decreases, and vice versa. Understanding this relationship is crucial for calculating the wavelength from the frequency or energy of the emitted photons.

Power and Photon Emission

Power (P) is defined as the rate at which energy is emitted or transferred. In this context, the power of the laser (20.0 mW) indicates how much energy is emitted per second. By knowing the total number of photons emitted and the time duration, one can calculate the energy per photon, which is essential for determining the wavelength of the laser light.

A laser produces 20.0 mW of red light. In 30.0 minutes, the laser emits 1.15×1020 photons. What is the wavelength of the laser?