Question

Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in \(\mathrm{nm}\) ) of such a proton moving at \(2.90 \times\) \(10^{8} \mathrm{~m} / \mathrm{s}\). (Mass of a proton \(\left.=1.673 \times 10^{-27} \mathrm{~kg} .\right)\)

Short answer

After following the steps, you found that the wavelength of the proton is approximately 0.00244 nm.

Step-by-step solution

Identify the given variables

Firstly, identify the given variables from the exercise. These would be the speed \(v = 2.90 \times 10^{8} m/s\) and the mass \(m = 1.673 \times 10^{-27} kg\) of a proton.

Use the de Broglie wavelength formula

The de Broglie wavelength formula is given by \(\lambda = \frac{h}{mv}\), where 'h' is Planck's constant, 'm' is the mass of the particle, and 'v' is the velocity of the particle. You need to substitute the given values into this equation to find the wavelength.

Substitute the given values

Now substitute the given values into the equation, remembering that Planck's constant is \(h = 6.63 \times 10^{-34} m^2 kg/s\). Doing so gives: \(\lambda = \frac{6.63 \times 10^{-34} m^2 kg/s}{(1.673 \times 10^{-27} kg)(2.90 \times 10^{8} m/s)}\).

Solve the equation

Solve the equation to find the value of \(\lambda\), the de Broglie wavelength of the proton. Remember you need to convert the final answer into nanometers (nm). So, 1 meter is equal to \(1 \times 10^{9}\) nanometers.

Check the answer

Lastly, check the answer to ensure it has a reasonable value based on the given conditions.

Key concepts

Understanding Protons

Protons are subatomic particles found in the nucleus of an atom. They are positively charged and, along with neutrons, make up most of the atom's mass. In fact, the mass of a proton is about \(1.673 \times 10^{-27} \ \text{kg}\).

• Protons play a crucial role in defining the identity of an element. The number of protons in an atom's nucleus is called the atomic number.
• Their positive charge equals the negative charge of electrons, making atoms electrically neutral.

When protons are accelerated to high speeds in a particle accelerator, they exhibit wave-like behavior. This phenomenon is perfectly illustrated by calculating their de Broglie wavelength, helping us bridge particle and wave physics.

The Role of Particle Accelerators

Particle accelerators are scientific machines designed to speed up and increase the energy of charged particles, like protons. They are used in various fields, from research in physics to medical applications.

• These machines use electric fields to accelerate particles and magnetic fields to steer them.
• In a particle accelerator, protons can be pushed to velocities close to the speed of light, greatly increasing their energy.

By studying accelerated protons, scientists can explore the fundamental forces of nature and simulate conditions similar to those shortly after the Big Bang. When protons move at high speeds, they exhibit wave properties described by their de Broglie wavelength.

Exploring Planck's Constant

Planck's constant is a fundamental constant in physics, denoted as \(h = 6.63 \times 10^{-34} \ \text{m}^2 \text{kg/s}\). It is a key element in the field of quantum mechanics, connecting the energy of photons to their frequency:
\[ E = hf \]

• This constant allows us to calculate the de Broglie wavelength, \(\lambda = \frac{h}{mv}\), which is pivotal in understanding the wave nature of particles.
• Planck's constant serves as a bridge between the macroscopic classical physics and the microscopic quantum world.

By using this constant, we see how even particles like protons have an associated wavelength when moving at high speeds, providing insights into their quantum behavior.