Refractometry

You may want to go back and revise the following:

• Refraction;

• Snell's Law;

• Critical Angle.

The associated equations are:

It's only in this course that the Physics code for refractive index is m (mu - a Greek letter 'm').  In most texts, it's n.  In this equation the refractive indices are absolute.  That means that the comparison is made with the refractive index of a vacuum, i.e. 1.0.  If we are comparing two different materials we write:

This is the relative refractive index as light passes from material of refractive index m1 to refractive index m2.

1m2 = m2 ÷ m1

For critical angle of light going from glass (refractive index 1.5) to water (refractive index 1.3):

gmw = 1.3 ÷ 1.5 = 0.866

The formula linking critical angle with refractive index is:

If we are going the other way, the equation is turned upside down:

2m1 = sin qc

gmw = 0.866

So:

sin qc = 0.866

So:

qc = sin^-1 0.866 = 60^o

With a refractometer, we have a set up like this:

The glass block is tilted until the black mark is just visible.  What we are trying to do is to work out the refractive index of the liquid.  We will go in the opposite directions to the arrows.

Let us assume that the glass has refractive index mg.  The refractive index of air is 1.0 (very nearly).  Therefore we can use Snell's Law to write:

mg sin q1 = 1 sin q2

So:

 sin q1 = sin q2

            mg

So:

mg = sin q2

        sin q1

Now:

qc = 90 - q1

So:

sin qc = sin 90 - sin q1 = 1 - sin q1 = cos q1

Since we are at the critical angle and going from an optically dense medium to a less dense, the relative refractive index is less than 1.

gml = ml ÷ mg

We need to say that:

gml = sin qc

Since the refractive index is the ratio of the speed of light in air to the speed of light in glass, we can write:

amg = c air

            c glass

Similarly:

gml = c glass

         c liquid

And:

aml = c air

            c liquid

We can combine these to give us an expression for aml.  If we multiply amg by gml we get:

c air ×    c glass = c air

 c glass     c liquid     c liquid

Therefore we can say that:

aml = amg × gml

And substituting:

aml = sin q2 × cos q1

        sin q1

Since sin/cos = tan, or cos/sin = 1/tan, we can finally write the expression:

aml = sin q2

        tan q1

And I have to say that this is one of the most tedious derivations I have done.  I think it would be an exceptional student who could do this derivation under examination conditions.

I think an easier approach is to:

    1. Apply Snell's law at the boundary of air and glass:

m air sin q2 = m glass sin q1

    2. Work out the critical angle by simply saying:

qc = 90 - q1

    3. Apply Snell's Law at the liquid - glass interface:

m liquid sin 90 = m glass sin qc

m liquid  = m glass sin qc

A graph of the refractive index and the sugar concentration is shown below:

So if you work out the refractive index, you can get the sugar concentration.  The recording of angles would be automated.

Polarimetry

If you look at a calculator through polaroid sunglasses, you may find that you cannot see the display.  This is because the two polarising filters are crossed.  Polarisation is a property of transverse waves.    Longitudinal waves are never polarised.  Light waves have an electrical component and a magnetic component at right angles.  Hence they are called electromagnetic waves.

My apologies for the wonky sine waves; not easy with the mouse.  We will use a model of a simple transverse wave.

Unpolarised light goes in any direction:

Polarised light goes in one plane only:

Light can be made to behave like this using a polarising filter or polaroid.

If we have two polaroids oriented in the same direction, the light passes through:

If the polaroids are crossed, we see this:

It's not just light waves that can be polarised.  Radio waves and other electromagnetic waves are easily polarised.

Sugar dissolved in water is optically active, which means that it can twist the polarised rays of light.  A polarimeter consists of a fixed polaroid and a polaroid that can be rotated.  The latter has a protractor scale.

The degree of polarisation depends on:

• the path length (i.e. the depth of the sample tube);
• the concentration of the sugar.

The graph of path length against rotation angle is a straight line:

The graph of rotation angle against concentration is also a straight line:

The specific rotation of light in an optically active solution is given by:

specific rotation = q

                              cL

The Physics codes are:

• q - the rotation in degrees;

• c - the concentration in g/ml;

• L - the length of the path in dm (1 dm = 0.1 m).

Notice again how SI units are not used.