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Calculate the saturation thermionic curr...

Calculate the saturation thermionic current if `120 W` is applied to a thoriated-tungsten filament of surface area `1.0 cm^2`. Assume that the surface radiates like a blackbody. The required constants are
`A = 3 xx10^4 A m^(-2) -K^2`, `varphi = 2.6 eV, k = 8.62 xx 10^(-5) eV kK^(-1)`
and `sigma = 6 xx 10^(-8) W m^(-2) K^(-4)`

Text Solution

Verified by Experts

The thermionic current is given by the Richardson-Dushman equation
`I = AST^2 e^(-(varphi)/(kT)` .. (i)
When the power input to the filament equals the power radiated, the temperature becomes constant. The thermionic current then becomes saturated. The power radiated is given by the Stefan's law
`P = SsigmaT^4`
or, `120 W = (1.0 xx 10^(-4) m^2) xx (6 xx 10^(-8 W m^(-2) K^(-4)) xx T^4`
or, `T=2114 K`.
Now `(varphi)/kT = (2.6 eV)/((8.62 xx 10^(-5 eV K^(-1)) (2114 K)) = 14.26`.
Putting in (i),
`i = 3 xx 10^4 A m^(-2) K^(-2) xx (1.0 xx 10^(-4) m^2) (2114 K)^2 e^(-14.26)`
`= (1.34 xx 10^7 A) e^(-14.26) = 8.6 A`.
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