Calculate the diffusion current density when the concentration of electron varies from the 1 * 1018 to 7 * 1017 cm-3 over a distance of 0.10 cm. D = 225cm2/sCorrect answer is '108A/cm2'. Can you explain this answer?

Question

Answer

Calculating the Diffusion Current Density:

Given:
- Concentration of electrons at x = 0: n1 = 1 * 10^18 cm^-3
- Concentration of electrons at x = 0.10 cm: n2 = 7 * 10^17 cm^-3
- Distance: d = 0.10 cm
- Diffusion coefficient: D = 225 cm^2/s

To calculate the diffusion current density, we can use Fick's first law of diffusion, which states that the diffusion current density (J) is proportional to the concentration gradient (dn/dx) and the diffusion coefficient (D).

Calculating the Concentration Gradient:

The concentration gradient (dn/dx) can be calculated by taking the difference in electron concentration (Δn = n2 - n1) and dividing it by the distance (Δx = d).

Δn = 7 * 10^17 cm^-3 - 1 * 10^18 cm^-3 = -3 * 10^17 cm^-3 (negative sign indicates a decrease in concentration)
Δx = 0.10 cm

The concentration gradient (dn/dx) is given by:
dn/dx = Δn/Δx = (-3 * 10^17 cm^-3) / (0.10 cm) = -3 * 10^18 cm^-4

Calculating the Diffusion Current Density:

Now, we can calculate the diffusion current density (J) using Fick's first law of diffusion:

J = -D * (dn/dx)

Substituting the values:
J = -225 cm^2/s * (-3 * 10^18 cm^-4)

Simplifying the expression, we get:
J = 675 * 10^18 cm^-2s^-1

To convert the current density from cm^-2s^-1 to A/cm^2, we need to multiply by the elementary charge (e = 1.6 * 10^-19 C) and divide by the area (A = 1 cm^2).

J = (675 * 10^18 cm^-2s^-1 * 1.6 * 10^-19 C) / (1 cm^2)

J ≈ 1.08 * 10^2 A/cm^2

Hence, the diffusion current density is approximately 108 A/cm^2.

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