This lecture is about light intensity and how to calculate it. Light intensity is the power
delivered by light per unit area of light, and it's a scalar value. The variable for
intensity is capital I and the units of intensity are watts per meter squared,
which makes sense because it's the amount of power being delivered per unit area, and the unit of
power is watts and the unit of area is meters squared. There are a few different equations
we use in IB physics to understand intensity. Besides just being equal to power over area
intensity is also proportional to the amplitude of a wave squared, and it's also proportional
to the inverse square of the distance from the light source. We're going to talk about what that
means in just a moment. First I'll do a quick example with the first equation. We can imagine
that light is delivering 500 joules of energy in 10 seconds to 10 square meters of surface area,
and we want to know the intensity of the light. So because power is energy over time, we can just
plug in our energy and our time to find that the power being delivered by the light is 50 watts,
and so the intensity is power over the area which is 50 watts over 10 meters squared, which is equal
to 5 watts per meter squared, so that would be the intensity of the light. Our understanding of
light intensity is based on some geometry, because when light leaves a source it spreads out in all
directions as a sphere. For example, we can look at this source of light here, this little blue dot
and we can imagine that there are rays of light leaving this dot in literally every direction.
That's what happens when light exits a light source, and because light always moves at the same
exact speed and it's moving in all directions, it's going to move out from the light source as a
sphere. As the sphere of light spreads out in all directions it's carrying the same amount of power,
but its area is increasing. This means that its intensity is decreasing as it spreads out. This
makes sense, if you look at the equation for intensity that I've written on the top right
you can see that the intensity of the light starts off being very large because the surface area that
we can imagine the light impacting is very small, but as the light begins to spread out there's more
surface area because the sphere is getting larger and larger, but the power being delivered by the
light remains the same, and so as a result the intensity shrinks. So the surface area
for a sphere is going to matter a lot for this topic. The area of a sphere is 4pi times the
radius of the sphere squared, so this will help us calculate the exact value of the intensity. Okay
let's do our first problem with intensity. When light is a distance X away from its source it
has an intensity of 40 watts per meter squared. What will be its intensity when it is 2x away
from its source? So first of all I'm going to write an equation for the original intensity,
and I knew that the original intensity is equal to the power delivered by the light over 4pi times R
squared and the radius here is the distance from the source, so that's going to be X. So the full
equation is power over 4 PI x squared. And we know that that's equal to 40 watts per meter squared.
The new intensity is going to be equal to that same amount of power because light maintains the
same amount of power as it spreads out divided by 4 pi times 2x squared because we're asking what is
the intensity when it is 2x away from its source. Multiplying that out gets me 4x squared because
I'm squaring both sides of that term, and for all proportional reasoning problems you want to
isolate the original equation in the new equation, so I'm going to isolate that original power over 4
PI x squared in this new equation, and when I draw out the other number I see that that's equal to
1/4 times the original equation, which means that it's going to be 1/4 times the original intensity,
which means that the new intensity is 1/4 over 40 which is equal to 10 watts per meter squared.
So if light has an intensity of 40 watts per meter squared at x distance from its source,
at 2x distance from its source it's going to have an intensity of 10 per meter squared. Okay this is
intensity example number two: when light is 20 meters away from its source it has an intensity
of 100 watts per meter squared. How far away from its source is the light when it has an intensity
of 70 watts per meter squared? So I'm going to write this out. I know that this is my intensity
equation and it's equal to 100 watts per meter squared. Solving for the power being delivered
by the light gets me 5.03 times 10 to the fifth watts. That's going to be useful for later in
the equation. So I know that the intensity is power over area and this power is remaining the
same no matter how far the light spreads out, and the area is always going to be equal to 4pi
times R squared, and we know that we're trying to figure out where this intensity is equal to
70 watts per meter squared. Isolating R gets me this equation which means that the radius,
the distance from the source to where the light is, is 23.9 meters. That's the exact location
where you'll get light with an intensity of 70 watts per meter squared. The intensity of light
is also directly proportional to the square of the light wave's amplitude. The proportional symbol,
which is a stretched out alpha, means that the intensity I is equal to some constant multiplied
by the amplitude squared, or A squared, so when you see this symbol, that stretched out alpha,
for proportionality, what that means is that the variable on the left is equal to the variable on
the right multiplied by some constant, but that constant can change depending on the situation.
So we can use this to solve problems. In example one a wave of light of amplitude 0.015 meters has
an intensity of 2.8 watts per meter squared. What would its intensity be if its amplitude is 0.03
meters? So solving this out, we know that this is our equation, we know that the intensity is equal
to some constant times a square. Plugging in the numbers for the original situation when it's at
the point zero one five meter amplitude, I can solve for the constant. So the constant in this
case is 12444 watts per to the negative 4th, and now I can take that constant and plug it into the
new situation where the amplitude is 0.03 meters instead, and when I do that I get a new intensity
of 11.2 watts per meter squared. In example number 2, how much does the amplitude of a light wave
have to increase for the intensity to increase to five times its original amount? So this time
I'm going to use the original and new notation. So I know that the original intensity is equal
to some constant times the original amplitude squared. Isolating the amplitude gets me this
equation. So I know that the new amplitude is equal to the square root of 5 times the original
intensity divided by that same constant C. So I want to isolate the original equation for the
original amplitude, and I do that by taking out the square root of five. And I know that
the original amplitude is equal to that square root of the intensity over C, so that means that
the new amplitude of light is going to be the square root of five multiplied by the original
amplitude. And that's the basic information you need to know about calculating light intensity.