Photon of frequency
u has a momentum associated with it. If c is the velocity of light, the moment

Q: Does a photon have mass?

A photon has zero *rest mass*. However, because it has energy and moves at the speed of light, it has an *effective relativistic mass* given by $m = E/c^2$.

Q: Can we use $p = mv$ for a photon?

Yes, if we use the effective mass $m = E/c^2$ and velocity $v = c$, we get $p = (E/c^2)c = E/c$.

Q: How is this related to the de Broglie wavelength?

The de Broglie wavelength is $\lambda = h/p$. For a photon, $\lambda = c/\nu$. Substituting this gives $p = h/\lambda = h\nu/c$.

Q: Why do we use $E=mc^2$ for a photon if it has no mass?

$E=mc^2$ applies to the total energy of a system. For a photon, its entire energy is kinetic, and it contributes to the momentum. The $m$ here is the relativistic mass equivalent of its energy.

Q: What is the difference between $h\nu$ and $hc/\lambda$?

They are exactly the same. Since the speed of light $c = \nu\lambda$, substituting $\nu = c/\lambda$ into $E = h\nu$ gives $E = hc/\lambda$.

Question

Photon of frequency $
u$ has a momentum associated with it. If $c$ is the velocity of light, the momentum is

Options:

(A) $\frac{
u}{c}$

(B) $h
u c$

(C) $\frac{h
u}{c^2}$

(D) $\frac{h
u}{c}$

Sigma Percentile · Accepted Answer

### The Dual Nature of Light
Light is a fascinating entity. Sometimes it behaves like a wave, and other times it acts like a stream of particles called **photons**. Even though a photon has absolutely zero rest mass, it carries energy and momentum. But how can something with no mass have momentum? Let's unravel this mystery.

### The Energy of a Photon
According to Max Planck's quantum theory, the energy of a single photon is directly proportional to its frequency. This is expressed by the famous equation:
$$E = h
u$$
where $h$ is Planck's constant and $
u$ is the frequency of the light.

### Einstein's Mass-Energy Equivalence
Albert Einstein showed us that mass and energy are two sides of the same coin through his iconic equation:
$$E = mc^2$$
Here, $m$ represents the relativistic or effective mass of the photon, and $c$ is the speed of light in a vacuum.

### Finding the Effective Mass
Since both equations describe the energy of the same photon, we can equate them:
$$mc^2 = h
u$$
By rearranging this equation, we can find the effective mass of the photon:
$$m = \frac{h
u}{c^2}$$
This tells us that while a photon has no rest mass, its energy gives it an effective mass when it's moving at the speed of light.

### Calculating the Momentum
In classical mechanics, momentum ($p$) is the product of mass and velocity. For a photon, its velocity is $c$. Therefore, its momentum is:
$$p = m 	imes c$$
Substituting the effective mass we just found:
$$p = \left(\frac{h
u}{c^2}ight) 	imes c$$
$$p = \frac{h
u}{c}$$

### The de Broglie Perspective
We can also arrive at this result using the de Broglie wavelength. The momentum of a particle is related to its wavelength by:
$$p = \frac{h}{\lambda}$$
Since the speed of a wave is the product of its frequency and wavelength ($c = 
u\lambda$), we can write $\lambda = \frac{c}{
u}$. Substituting this into the de Broglie equation gives:
$$p = \frac{h}{\frac{c}{
u}} = \frac{h
u}{c}$$
Both methods beautifully converge to the same result, proving the profound consistency of modern physics!

Animated Solution for Physics - Dual Nature of Matter and Radiation: Photon of frequency $ν$ has a momentum associated with it. If $c$ is the velocity of light, the momentum is

Select Answer:

Visualized Solution

$E = h ν$

$E = m c^{2}$

$m c^{2} = h ν$

$m = \frac{h ν}{c ^{2}}$

$p = m c$

$p = \frac{h ν}{c}$

$p = \frac{h}{λ}$

The Sigma Insight: Photon Theory of Light

The Dual Nature of Light Light is a fascinating entity

The Energy of a Photon According to Max Planck's quantum theory, the energy of a single photon is directly proportional to its frequency

Einstein's Mass-Energy Equivalence

Finding the Effective Mass

Calculating the Momentum In classical mechanics, momentum ( $p$ ) is the product of mass and velocity

The de Broglie Perspective We can also arrive at this result using the de Broglie wavelength

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Animated Solution for Physics - Dual Nature of Matter and Radiation: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Photon of frequency ν has a momentum associated with it. If c is the velocity of light, the momentum is