$($For coding in Python$)$ We can write the Planck function as: B$\backslash$lambda $(\backslash$lambda $,$ T$)=2$hc$2\backslash$lambda $5($e hc $\backslash$lambda kT $1)$

Wien$$s Law is an expression indicating the wavelength where this blackbody emission peaks $($is highest$)$ for a given temperature: $\backslash$lambda max $=$ b T $(2)$ where $\backslash$lambda max is the peak wavelength in meters $($m$),$ T is the blackbody temperature in Kelvin $($K$),$ and b is Wien$$s displacement constant equal to $2\mathrm{.}898\backslash$times $103$ m $$ K$.$

write code that numerically determines the wavelength where the peak of the Planck function occurs for a given temperature.

compare your results to the Wien$$s Law expression. Equation $1$ above will be used in the loop to find wavelength where the blackbody spectrum has its maxiumum value; Use Equation $2$ to check the wavelength that your loop identifies. The numerical solution will give you an approximate solution while Wien$$s law gives an exact solution. While writing code, you should use Wien$$s law to get an idea of the approximate values the numerical code should give you.

Your program must contain the following:

$$ A variable that represents the blackbody temperature, so your peak$-$finding algorithm can be applied to different blackbodies by modifying a single line of code $($i$.$e$.$ the line where the value of that temperature variable is set$).$

$$ A variable representing the initial wavelength guess.

$$ A variable representing the wavelength increment, or step size.

$$ A variable representing the current value of B$.$

$$ A variable representing the previous value of B $($this could be zero at start of loop$).$

$$ A variable that keeps track ofstep number.

$$ Lists to store the wavelengths and B values. Here$$s a sketch of the procedure the code should follow to numerically identify the peak of the blackbody function. the code should:

$1.$ Calculate the intensity of light B$\backslash$lambda $(\backslash$lambda $,$ T$)$ for a given wavlength $(\backslash$lambda $)$ and blackbody temperature $($T$)$

$$ Confirm the calculation of B does the math correctly. For a wavelength of $\backslash$lambda $=700$nm $($red light$)$ and a temperature of T $=6000$K $($temperature of the Sun$),$ your calculation should yield $2\mathrm{.}38\backslash$times $1013$J$/$s$/$sr$/$m$3$

$2.$ Be able to start with an initial guess for a wavelength that is close to $($but less than$)$ the peak blackbody wavelength.

$3.$ Use while loops to increment the guess for wavelength and recompute B$\backslash$lambda until finding the $$turnover point,$$ the point where the value of B$\backslash$lambda goes from increasing to decreasing.

$-$ Don$$t use Wien$$s law to predetermine the peak. The point is to find the peak numerically!

$4.$ Store wavelength and B values in a list to help verify that the algorithm finds the peak.

$5.$ Use a compound conditional statement $($Booleans$!)$ to prevent the while loop from taking more than $1000$ steps

Answer each question below in its own cell using a combination of code cells and Markdown cells. It$$s fine to write code to print out the explanation of the answers

$1.$ Using your numerical algorithm, what is the wavelength of peak brightness of a T $=12,000$ K blackbody in nanometres? Use $10$ nm as the initial guess and a step$-$size of $+20$ nm$.$

$2.$ How many iterations did it take for the algorithm to find the blackbody peak?

$3.$ How close is the result to the analytical result $($Wien$$s Law$)?$

$4.$ To confirm, print your list$($s$)$ of wavelength and B values. Use a for loop to print one pair of wavelength and brightness values per line. Make sure the wavelength prints in nanometers with the number taking up at least $5$ characters, and with $0$ digits after decimal place. Make sure the blackbody emission prints in fixed$-$point notation with the number taking up at least $20$ characters and with $1$ digit after the decimal place.

$5.$ In a new code cell, copy and re$-$run the algorithm for the same temperature, with a step$-$size of $+1$ nm$.$ How many iterations did it take to find the peak? How close is the result to Wien$$s Law now? print your list$($s$)$ of wavelength and B values.