Calculate the Ratio of Real Depth to Apparent Depth for a Liquid with Refractive Index 1.5

Correct Answer: 1.5 : 1

Understanding Real Depth and Apparent Depth

When an object placed inside a transparent liquid is viewed from above through air, the object appears to be closer to the surface than it actually is. This optical effect occurs because light rays change their direction when they travel from the liquid into air.

The actual vertical distance of the object below the liquid surface is called the real depth. The depth at which the object appears to be located when viewed from air is called the apparent depth.

Since the object appears raised toward the surface, the apparent depth is smaller than the real depth. The relationship between these two depths is directly connected to the refractive index of the liquid.

Relationship Between Refractive Index, Real Depth, and Apparent Depth

When an object inside a liquid is viewed approximately normally from air, the refractive index of the liquid relative to air is given by:

Refractive index = Real depth / Apparent depth

Using symbols:

μ = Real depth / Apparent depth

Here, μ represents the refractive index of the liquid relative to air.

This relation shows that the ratio of real depth to apparent depth is numerically equal to the refractive index of the liquid.

Given Value in the Question

The refractive index of the liquid relative to air is:

μ = 1.5

We have to calculate:

Real depth / Apparent depth

Step-by-Step Calculation of the Required Ratio

Step 1: Apply the Apparent Depth Formula

The relationship between refractive index and depth is:

μ = Real depth / Apparent depth

Step 2: Substitute the Given Refractive Index

Substituting μ = 1.5:

1.5 = Real depth / Apparent depth

Therefore:

Real depth / Apparent depth = 1.5

Step 3: Express the Answer as a Ratio

The decimal value 1.5 can be written as:

1.5 = 1.5/1

Therefore, the required ratio is:

Real depth : Apparent depth = 1.5 : 1

The same ratio may also be written using whole numbers as:

Real depth : Apparent depth = 3 : 2

Why Does an Object Inside a Liquid Appear Raised?

Light from an object inside the liquid travels from an optically denser medium, the liquid, into an optically rarer medium, air. As the light rays enter air, they bend away from the normal because of refraction.

The human eye assumes that light travels in straight lines. Therefore, the refracted rays entering the eye are mentally traced backward in straight lines. These backward extensions appear to originate from a point above the actual position of the object.

As a result, the object appears closer to the liquid surface than it really is. This is why the apparent depth is less than the real depth.

Physical Meaning of the Refractive Index Value 1.5

A refractive index of 1.5 means that the real depth is 1.5 times the apparent depth. For example, if the apparent depth of an object is 10 cm, its real depth would be:

Real depth = 1.5 × 10 = 15 cm

Similarly, an object located at a real depth of 15 cm would appear to be only 10 cm below the surface when viewed from air under the conditions for which the apparent-depth relation is applicable.

Alternative Form of the Apparent Depth Formula

The relation can also be rearranged to calculate apparent depth directly:

Apparent depth = Real depth / μ

For μ = 1.5:

Apparent depth = Real depth / 1.5

This confirms that the apparent depth is smaller than the real depth, as expected when an object in a denser medium is viewed from a rarer medium.

Final Answer

The ratio of the real depth to the apparent depth is 1.5 : 1.

Equivalently, the ratio can be expressed as:

Real depth : Apparent depth = 3 : 2

This follows directly from the relation μ = Real depth / Apparent depth. Since the refractive index of the liquid relative to air is 1.5, the required ratio is 1.5 : 1.