Glass Thickness for Light Absorption: 90%, 99%, 99.9%

[Astro Student]

Homework Statement

A slab of glass 0.5 m thick absorbs 50% of light passing through it. Compute how thick of a slab of identical glass you would need that:
- absorbs 90% of light
- absorbs 99% of light
- absorbs 99.9% of light

Homework Equations

I spent an hour looking for formulas in my textbook and couldn't find any, which is the primary reason I'm having trouble.

The Attempt at a Solution

N/A

 

[blue_leaf77]

The simplest mathematical formulation to describe light absorption is through the equation $I(z) = e^{-\alpha z} I(0)$, where $z$ the distance traveled and $\alpha$ an absorption coefficient.

 

[Mister T]

I typed light absorption as a function of glass thickness into the Google search engine and it came up on the first hit.

 

[Astro Student]

The simplest mathematical formulation to describe light absorption is through the equation $I(z) = e^{-\alpha z} I(0)$, where $z$ the distance traveled and $\alpha$ an absorption coefficient.

So the quantity that we are trying to measure here is intensity? I do see a formula in my textbook saying I(z) = I₀e^-τ(x). Could that be it?

EDIT: This actually appears to be a formula for intensity through a medium of gas. Nevermind.

 

[blue_leaf77]

I(z) = I0e-τ(x)

Shouldn't the exponent be a function of z?

 

[Astro Student]

Shouldn't the exponent be a function of z?

Yes, I mistyped it. Both I and tau should be functions of x according to my textbook. I'm still not finding a formula for absorption of light by a solid anywhere in my textbook, only for gases.

 

[blue_leaf77]

If you can't find it in your book, you should look for it somewhere else, shouldn't you?

 

[Astro Student]

Another formula in my textbook says ΔI/Δ = -(nSΔx)σ/S = -nσΔx. σ here is the cross-section, Δx the thickness, and n the number density. For the sake of this problem, the number density and cross-section would remain constant with Δx as the only independent variable to change ΔI/I. Would this be an acceptable solution to this problem?

 

[blue_leaf77]

ΔI/Δ = -(nSΔx)σ/S = -nσΔx.

This is accurate only when the thickness and/or the absorption coefficient are small. Look what happen if you make the deltas infinitesimal ($\Delta \rightarrow d$),
$$
\frac{dI}{I} = n\sigma dx
$$
and then integrate both sides. What expression will you get?

 

[Mister T]

A slab of glass 0.5 m thick absorbs 50% of light passing through it.

What would two of these slabs do?

I spent an hour looking for formulas in my textbook and couldn't find any, which is the primary reason I'm having trouble.

Perhaps you underestimate your own powers of reason.

 

[Astro Student]

This is accurate only when the thickness and/or the absorption coefficient are small. Look what happen if you make the deltas infinitesimal ($\Delta \rightarrow d$),
$$
\frac{dI}{I} = n\sigma dx
$$
and then integrate both sides. What expression will you get?

lnI = -nσ dx + C₁.
If I use the knowledge that at x = 0 I = I₀ we can use differential equations (I worked them out on a sheet of paper) to get C₁ = lnI₀ and then I(x) = I₀e^-nσx.

I think...

 

[blue_leaf77]

lnI = -nσ dx + C₁.
If I use the knowledge that at x = 0 I = I₀ we can use differential equations (I worked them out on a sheet of paper) to get C₁ = lnI₀ and then I(x) = I₀e^-nσx.

I think...

Yes, that's right, and how does it compare to the equation in comment #2?

 

[Astro Student]

Yes, that's right, and how does it compare to the equation in comment #2?

It's looks similar. In the equation in comment 2 I would assume number density and cross section were combined into a single constant.

 

[blue_leaf77]

So, what is still halting you to answer your homework questions?

 

[Astro Student]

So, what is still halting you to answer your homework questions?

Nothing anymore. Thank you very much!