A particle is travelling 4 times as fast as an electron. Assuming the ratio of de-Broglie wavelength

Q: What exactly is the de-Broglie wavelength?

It's the wavelength associated with a moving particle, given by $\lambda = \frac{h}{p}$, where $p$ is the particle's momentum. It shows that matter has wave-like properties.

Q: Why do we use ratios to solve this problem?

Ratio problems are common in physics because they allow us to cancel out constants (like Planck's constant $h$) and focus only on the variables that change, making the math much simpler.

Q: Does a faster particle always have a shorter wavelength?

Yes, if the mass is constant, wavelength is inversely proportional to velocity. However, in this problem, both mass and velocity are different for the two particles, which is why the faster particle actually has a longer wavelength here.

Q: Can we use kinetic energy instead of velocity in the formula?

Yes, you can use the alternate form $\lambda = \frac{h}{\sqrt{2mK}}$. But since the problem explicitly gives us the relationship between their velocities, using $\lambda = \frac{h}{mv}$ is much more direct.

Q: What if the mystery particle was a proton?

A proton has a known mass (about $1836$ times that of an electron). If it were a proton, the wavelength and velocity ratios would have to be completely different to satisfy the equation. Here, we are finding an unknown mass relative to the electron.

Question

A particle is travelling $4$ times as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is $2 : 1$, the mass of the particle is

Options:

(A) $\frac{1}{16}$ times the mass of electron

(B) $8$ times the mass of electron

(C) $16$ times the mass of electron

(D) $\frac{1}{8}$ times the mass of electron

Sigma Percentile · Accepted Answer

### Racing Particles and the Magic of de-Broglie Wavelengths

Imagine a race between an electron and a mystery particle. We are given some intriguing clues about their speeds and the invisible "matter waves" they create as they move. Our mission? To uncover the mass of this mystery particle.

### Analyzing the Setup

The problem provides us with two critical pieces of information. First, the mystery particle is moving incredibly fast—exactly four times as fast as the electron. Mathematically, we can write this as:

$$v_p = 4v_e$$

Second, we are given the ratio of their de-Broglie wavelengths. The particle's wavelength is twice that of the electron:

$$\frac{\lambda_p}{\lambda_e} = \frac{2}{1}$$

### The Master Equation

To connect mass, velocity, and wavelength, we must summon the famous de-Broglie equation. Louis de Broglie proposed that every moving particle has an associated wave, and its wavelength $\lambda$ is inversely proportional to its momentum ($p = mv$). The equation is:

$$\lambda = \frac{h}{mv}$$

where $h$ is Planck's constant.

### The Power of Ratios

In physics, when comparing two states or two particles, taking a ratio is a superpower. It elegantly eliminates constants. Let's divide the de-Broglie wavelength of the particle by that of the electron:

$$\frac{\lambda_p}{\lambda_e} = \frac{\frac{h}{m_p v_p}}{\frac{h}{m_e v_e}}$$

Notice how Planck's constant $h$ beautifully cancels out! We are left with an inverse ratio of their momentums:

$$\frac{\lambda_p}{\lambda_e} = \frac{m_e v_e}{m_p v_p}$$

### Final Calculation

Now, we simply substitute the values we extracted from the problem statement. We know the wavelength ratio is $2$, and we can replace $v_p$ with $4v_e$:

$$2 = \left(\frac{m_e}{m_p}\right) \cdot \left(\frac{v_e}{4v_e}\right)$$

The velocity of the electron, $v_e$, cancels out, leaving us with a straightforward algebraic equation:

$$2 = \left(\frac{m_e}{m_p}\right) \cdot \frac{1}{4}$$

Multiplying both sides by $4$, we get:

$$8 = \frac{m_e}{m_p}$$

Rearranging to solve for the mass of the mystery particle, $m_p$, we find:

**$$m_p = \frac{1}{8} m_e$$**

The mass of the particle is exactly one-eighth the mass of the electron. This elegant cancellation is why ratio problems are so satisfying to solve!

Animated Solution for Physics - Dual Nature of Matter and Radiation: A particle is travelling $4$ times as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is $2 : 1$ , the mass of the particle is

Select Answer:

Visualized Solution

Visualizing the Setup

Velocity Relation

Wavelength Relation

The de-Broglie Equation

Setting up the Ratio

Substituting Values

Solving for Mass

Conclusion

The Sigma Insight: Matter Waves and de Broglie Relation

Racing Particles and the Magic of de-Broglie Wavelengths

Analyzing the Setup

The Master Equation

The Power of Ratios

Final Calculation

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