There is an infinite non-multiplying medium with diffusion length 3 cm and diffusion coefficient 1 cm. There are 3 point sources of neutrons on the x-y plane:
„h Source S1 = 108 s-1 at (x, y) = (2 cm, 0)
„h Source S2 = 2*108 s-1 at (x, y) = (0, 2 cm)
„h Source S3 = 4*109 s-1 at the origin.
At the point (x, y) = (2 cm, 4 cm), find:
(a) the total neutron flux
(b) the magnitude and direction of the total neutron current.
Thanks
To find the total neutron flux at the point (x, y) = (2 cm, 4 cm), we need to consider the contribution from each of the three point sources.
(a) The total neutron flux is the sum of the fluxes from each source. The flux from a point source can be calculated using the formula:
Flux = (Source strength) / (4πD²)
where Source strength is the rate at which neutrons are emitted from the source, and D is the distance from the source to the point of interest (in this case, (2 cm, 4 cm)).
Let's calculate the flux from each source:
Flux1 = (S1) / (4πD1²), where S1 = 10^8 s^-1 and D1 is the distance between (2 cm, 0) and (2 cm, 4 cm). Using the distance formula, D1 = sqrt((2-2)^2 + (4-0)^2) = 4 cm.
Flux1 = 10^8 / (4π * 4^2) = 10^8 / (64π) = 10^8 / 201.06 ≈ 4.97 * 10^5 s^-1 cm^-2
Similarly:
Flux2 = (S2) / (4πD2²), where S2 = 2 * 10^8 s^-1 and D2 is the distance between (0, 2 cm) and (2 cm, 4 cm). Using the distance formula, D2 = sqrt((2-0)^2 + (4-2)^2) = 2√2 cm.
Flux2 = 2 * 10^8 / (4π * (2√2)^2) = 2 * 10^8 / (16π) ≈ 3.98 * 10^6 s^-1 cm^-2
Flux3 = (S3) / (4πD3²), where S3 = 4 * 10^9 s^-1 and D3 is the distance between the origin and (2 cm, 4 cm). Using the distance formula, D3 = sqrt((2-0)^2 + (4-0)^2) = 2√5 cm.
Flux3 = 4 * 10^9 / (4π * (2√5)^2) = 4 * 10^9 / (80π) ≈ 1.59 * 10^7 s^-1 cm^-2
Now, we can find the total neutron flux:
Total Flux = Flux1 + Flux2 + Flux3 ≈ 4.97 * 10^5 + 3.98 * 10^6 + 1.59 * 10^7 ≈ 2.09 * 10^7 s^-1 cm^-2
Therefore, the total neutron flux at the point (2 cm, 4 cm) is approximately 2.09 * 10^7 s^-1 cm^-2.
(b) To find the magnitude and direction of the total neutron current at the point (2 cm, 4 cm), we need to consider both the flux (neutron density) and the diffusion coefficient.
The neutron current density vector J is given by:
J = -D * ∇φ
where D is the diffusion coefficient and ∇φ is the gradient of the neutron flux φ.
In this case, since the medium is infinite and non-multiplying, the neutron flux does not depend on position. Therefore, the gradient of the neutron flux (∇φ) is zero, and the neutron current density J is also zero.
Hence, the magnitude of the total neutron current at the point (2 cm, 4 cm) is zero, and the direction is undefined.