Electrons with a speed of 1.1×10^6 m/s pass through a double-slit apparatus. Interference fringes are detected with a fringe spacing of 2.0 mm.
What will the fringe spacing be if the electrons are replaced by neutrons with the same speed? (in μm)
Well, let me tell you, those electrons must be booking it if they're going at 1.1×10^6 m/s! But now we're talking about neutrons. They might be a bit slower, but they can still put on a good show.
When it comes to interference fringes, the spacing depends on the wavelength of the particles. Since neutrons are particles too (and not just clown friends), they have a wavelength associated with them. And guess what? It turns out that the wavelength of a particle is inversely proportional to its speed.
So, if the electrons had a certain fringe spacing, when we replace them with neutrons of the same speed, the fringe spacing will change because the wavelength of neutrons will be different.
But don't worry, I won't leave you hanging. The fringe spacing for the neutrons can be found using the same formula as for the electrons:
Fringe spacing = (wavelength × distance)/slit separation
Since the speed of the neutrons is the same as the speed of the electrons, you can use the same speed value to calculate the new wavelength. Once you have the new wavelength, you can find the new fringe spacing by plugging it into the formula above.
But hey, I know this is a technical question, so let me put it in a way that might make you chuckle: "Why did the neutron go to the double-slit party? To see if it could make the fringes even more fashionable!"
Hope that helps!
To find the fringe spacing for neutrons, we need to use the de Broglie wavelength equation:
λ = h / p
Where:
λ is the wavelength,
h is the Planck's constant (6.626 x 10^-34 J·s),
p is the momentum.
The momentum of a particle is given by:
p = mv
Where:
m is the mass of the particle,
v is the speed of the particle.
Given that electrons and neutrons have the same speed, the ratio of their wavelengths will be equal to the ratio of their masses:
λ_neutron / λ_electron = m_electron / m_neutron
We can rearrange and solve this equation for the wavelength of neutrons:
λ_neutron = (m_electron / m_neutron) * λ_electron
First, let's find the wavelength of electrons:
Given speed of electrons (v_electron) = 1.1×10^6 m/s
Let's assume the mass of an electron (m_electron) = 9.109 x 10^-31 kg
Using the momentum equation:
p_electron = m_electron * v_electron
p_electron = (9.109 x 10^-31 kg) * (1.1×10^6 m/s)
p_electron = 9.109 x 10^-25 kg·m/s
Now, let's find the wavelength of electrons using the de Broglie equation:
λ_electron = h / p_electron
λ_electron = (6.626 x 10^-34 J·s) / (9.109 x 10^-25 kg·m/s)
λ_electron ≈ 7.25 x 10^-10 m
Now, let's find the mass of neutrons:
Let's assume the mass of a neutron (m_neutron) = 1.675 x 10^-27 kg
Now, we can calculate the wavelength of neutrons:
λ_neutron = (m_electron / m_neutron) * λ_electron
λ_neutron = (9.109 x 10^-31 kg) / (1.675 x 10^-27 kg) * (7.25 x 10^-10 m)
λ_neutron ≈ 3.93 x 10^-12 m
Now, to find the fringe spacing for neutrons, we can use the following equation:
Fringe spacing = wavelength * distance between slits
Given that the distance between the slits is 2.0 mm or 2 x 10^-3 m, we can calculate the fringe spacing for neutrons:
Fringe spacing_neutron ≈ (3.93 x 10^-12 m) * (2 x 10^-3 m)
Fringe spacing_neutron ≈ 7.86 x 10^-15 m
Converting to micrometers (μm):
Fringe spacing_neutron ≈ 7.86 x 10^-6 μm
Therefore, the fringe spacing for neutrons will be approximately 7.86 μm.
To determine the fringe spacing when replacing electrons with neutrons, we need to understand the relationship between fringe spacing, wavelength, and the distance between the double slits.
Let's start by finding the wavelength of electrons with a speed of 1.1×10^6 m/s using the de Broglie wavelength equation:
λ = h / p
where λ is the wavelength, h is Planck's constant (6.626 × 10^-34 J·s), and p is the momentum of the particle.
To find the momentum of an electron, we can use the equation:
p = m * v
where p is the momentum, m is the mass of the particle, and v is the velocity.
The mass of an electron (m) is approximately 9.10938356 × 10^-31 kg.
Plugging in the values:
p = (9.10938356 × 10^-31 kg) * (1.1×10^6 m/s)
p = 1.002 × 10^-24 kg·m/s
Now, we can find the wavelength:
λ = (6.626 × 10^-34 J·s) / (1.002 × 10^-24 kg·m/s)
λ ≈ 6.600 × 10^-11 m
Next, we can find the fringe spacing using the following formula:
s = λ * L / d
where s is the fringe spacing, λ is the wavelength, L is the distance from the double slits to the screen, and d is the distance between the double slits.
In this case, we assume L and d remain unchanged.
Given that the fringe spacing for electrons is 2.0 mm (or 2.0 × 10^-3 m), we can rearrange the formula to solve for s:
s = (λ * L) / d
Now let's substitute the known values into the equation:
s = (6.600 × 10^-11 m) * (L) / (d)
Since L and d remain the same, the fringe spacing (s) for neutrons with the same speed will be the same as for the electrons, which is 2.0 × 10^-3 m (or 2.0 μm).
Therefore, the fringe spacing for neutrons will also be 2.0 μm.