How do you find the approximate interval that displays the solution?

T = 350; %in K
epsi = 0.9; %emissitivity
c = 3e8; %speed of light
kb = 1.38e-23; %boltzman constant
h = 6.63e-34; %plancks constant

%full expression
Tb = (h*c)./(lambda*kb.*log(1+(1./epsi)*(exp((h*c)./(lambda*kb*T))-1)));

%Approximate expression
Tb_approx = epsi*T;

I need to find the approximate solution for both of these equations but I don't know how. Is there a way to plot both functions to find the interval results are displayed on?

[Ted Shultz]

What are you tring to solve? the approimate functions as a function of lambda?

[Ted Shultz]

As a side note, you may want to use:

kb = physconst('boltzman');
c = physconst('lightspeed');

[Isabel Maczka]

I need to find when the approximate solution is valid but there is no interval given

[Walter Roberson]

pysconst() appears to require the Phased Array System Toolbox

[Walter Roberson]

Tb = @(lambda) (h*c)./(lambda*kb.*log(1+(1./epsi)*(exp((h*c)./(lambda*kb*T))-1)));
fzero(@(lambda) Tb(lambda) - Tb_approx, rand())

The solution is about 3.20966703379108e+16

Answer 1

[David Goodmanson]

Hello Isabel,

I believe the idea here is that when lambda is large enough, the expression for Tb gets increasingly close to Tb_approx. So the question is, how large does lambda have to be so that Tb_approx is good? There is no hard and fast answer to that, but you can set up an array for lambda, plot Tb as a function of lambda, and compare to Tb_approx.

As far as the domain of lambda, there is a way to find a lambda0 that sets the scale of this problem. In the Tb expression, take a look at the Bose-Einstein factor in the denominator:

(exp((h*c)./(lambda*kb*T)) - 1)

A characteristic lambda0 occurs when the argument of the exponential is 1:

(h*c) / (lambda0*kb*T)   = 1   -->   lambda0 = (h*c) / (kb*T)  = 4.12e-5.

You will have to go significantly above that to get close to Tb_approx. You can do something like

lambda = logspace(-7,-1,1000);
< calculate Tb here > 
Tb_approxvec = Tb_approx*ones(size(lambda));   % horiz line
semilogx(lambda,Tb,lambda,Tb_approxvec);       % log scale for lambda is appropriate
ylim([300 400])

It's interesting that for small lambda, the expession goes over to T.