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The Child-Langmuir Law describes the relation between the current and the voltage between the cathode and the anode. Generally the equation for Child-Langmuir Law is J=4/9*K/s*V^(3/2)=2.330*10^(-6)*V^(3/2)/s while J is the current (A) V is the voltage between the cathode and the anode (V) K = eps0*sqrt(2*e/m) is a constant, e is the charge of electron, m is the mass of electron, eps0 is the permittivity of vacuum s is a constant depending on the shape of the anode and cathode, its unit is cm^2. For simplicity, consider a 1-D problem, i.e. the anode and the cathode are two infinite parallel plates, one at x=0, the other at x=d. In this case the shape parameter s in the Child-Langmuir Law is s=d^2 Now let's derive this equation. Assume the potential is V(x), we'll have Possion's equation d^2 V rho ----- = - ---- d^2 x eps while rho(x) is the density of the charge, eps is the permittivity of the medium. Assume V(0)=0, electrons move from x=0 toward x=d, at x=x' electron will gain energy e*V(x'), therefore its speed is v = sqrt(2*e*V/m) while m is the mass of the electron Assume the current distribution is J(x). It should be noticed that in steady state J is constant due to the conservation of electric charges: d rho d J ----- = - ------ = 0 d t d x The relation between rho(x) and J is J = rho * v Therefore rho = J/v = J/sqrt(2*e*V/m) By substitution of the above equation into Possion's equation, we can obtain d^2 V J/sqrt(2*e*V/m) ----- = - ---------------- d^2 x eps This ordinary differential equation can be easily solved by assuming V(x)=a*x^b and substituting it into the above equation J/sqrt(2*e/m) a*b*(b-1)*x^(b-2) = - ---------------- /sqrt(a*x^b) eps therefore b-2=-b/2 a^(3/2)*b*(b-1)=-J/sqrt(2*e/m)/eps From the above equations we obtain b=4/3 a=[-J/sqrt(2*e/m)/eps*9/4]^(2/3) Therefore V(x)=[-J/sqrt(2*e/m)/eps*9/4]^(2/3)*x^(4/3) which can be simplified as J=-4/9/x^2*eps*sqrt(2*e/m)*V^(3/2) If V(x)>0, J<0, which means the current flows from higher potential to lower potential, just in inverse direction of the motion of the electrons. References J. P. Barbour et al., Space-charge effects in field emission, Phys. Rev. 92 (1): 45-51, 1953.

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