Spherical Refraction Formula

1. Visualizing Refraction at a Spherical Surface

Imagine a ray of light originating from an object O in a medium with refractive index $n_1$. It strikes a curved boundary (part of a sphere) and enters a second medium with refractive index $n_2$, forming an image I.

To solve any problem involving this setup, we must first master the Cartesian Sign Convention. Getting the signs right is more than half the battle.

To summarize the crucial rules:

1. Origin: The pole (P) of the spherical surface is the origin for all measurements.
2. Direction: The direction of the incident light is taken as positive. All distances measured in this direction are positive, and all distances measured opposite to it are negative.
3. Radius of Curvature (R):
   • If the center of curvature C is on the positive side (as in the diagram above, for a convex surface), R is positive.
   • If the center of curvature C is on the negative side (for a concave surface), R is negative.

2. Deriving the Spherical Surface Formula

With the sign convention established, we can now derive the mathematical relationship between $u,v,R,n_1,$ and $n_2$. The derivation is a beautiful application of Snell's Law combined with some geometry.

The derivation in the video leads us to the fundamental equation for refraction at a single spherical surface:

$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$

Let's break down the terms based on the helpful mnemonic from the video:

• $\frac{n_2}{v}$: $n_2$ is the refractive index of the medium where the refracted ray (and image) is. $v$ is the image distance.
• $\frac{n_1}{u}$: $n_1$ is the refractive index of the medium where the incident ray (and object) is. $u$ is the object distance.
• $\frac{n_2-n_1}{R}$: The difference in refractive indices divided by the radius of curvature.

This formula is universally applicable for any spherical surface (convex or concave) and for light traveling from rarer-to-denser or denser-to-rarer media, as long as you apply the sign convention correctly.

3. Applying the Formula: Problem Solving

Now, let's put this formula into practice. This is where your NEET preparation gets a vigorous workout.

Example 1: A Straightforward Application

Let's start with a classic problem.

Question: Light from a point source in air ($n_1=1$) falls on a spherical glass surface ($n_2=1.5$, radius of curvature $R=+20$ cm). The distance of the light source from the glass surface is 100 cm. At what position is the image formed? (This is Example 9.6 from the NEETprep resource LINK).

Solution Steps:

1. List the knowns with signs:

   • The object is in air: $n_1=1$.
   • The image is in glass: $n_2=1.5$.
   • Object distance: Light travels from the object to the surface. We measure from the pole P back to the object, which is against the direction of incident light. So, $u=-100$ cm.
   • Radius of curvature: The surface is convex towards the object, so its center of curvature is inside the glass, in the positive direction. So, $R=+20$ cm.
   • Image distance: $v=?$

2. Apply the formula:
   $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$
   $\frac{1.5}{v}-\frac{1}{-100}=\frac{1.5-1}{+20}$

3. Solve for v:
   $\frac{1.5}{v}+\frac{1}{100}=\frac{0.5}{20}=\frac{1}{40}$
   $\frac{1.5}{v}=\frac{1}{40}-\frac{1}{100}$
   To find a common denominator (200):
   $\frac{1.5}{v}=\frac{5}{200}-\frac{2}{200}=\frac{3}{200}$
   $v=\frac{1.5\times 200}{3}=\frac{300}{3}=100\text{ cm}$

The positive sign for $v$ indicates that the image is formed at 100 cm inside the glass, in the same direction as the incident light. This is a real image.

Example 2: Multi-Surface System (A Glass Sphere)

NEET problems often involve more than one refracting surface. The following example is crucial as it demonstrates the method for handling such systems.

Key takeaways from the glass sphere problem:

• Step 1: Refraction at the First Surface. You apply the formula just as in our first example to find the position of the first image, $I_1$. In the video, this was found to be at $v_1=+90$ cm.
• Step 2: The Bridge Concept. The image $I_1$ formed by the first surface becomes the object for the second surface.
• Step 3: Refraction at the Second Surface.
  • Re-evaluate everything: The pole is now the pole of the second surface. The incident medium is now glass ($n_1=1.5$) and the refracted medium is air ($n_2=1$). The direction of incident light remains the same.
  • New Object Distance: The new object distance, $u_2$, is the distance from the second pole to $I_1$. In the video, since $I_1$ was at +90 cm from the first pole and the sphere's diameter was 20 cm, the new object distance was $u_2=90-20=+70$ cm. Note that $u_2$ is positive because this object is on the positive side of the new pole. This is a virtual object.
  • New Radius of Curvature: The center of curvature for the second surface is on the negative side of its pole, so R is negative.
  • Apply the formula again with these new values to find the final image position.

This method of treating the image from one surface as the object for the next is the fundamental principle you will use later for analyzing thick lenses and combinations of lenses.

Conclusion

In this lesson, we have successfully bridged the gap between plane and curved surfaces. You have learned how to derive and, more importantly, apply the formula for refraction at a single spherical surface.

Key Takeaways:

• The governing formula is $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$.
• Correctly applying the Cartesian sign convention is non-negotiable for success. The direction of incident light is always positive.
• For systems with multiple surfaces (like a sphere or a lens), the image formed by the first surface serves as the object for the second. You must re-evaluate all distances from the new pole for each subsequent refraction.

Preview of the Next Lesson:
Before we combine two spherical surfaces to officially form lenses, we will take a slight but important detour to study refraction through prisms. Analyzing prisms will deepen your understanding of how a material's refractive index and the geometry of its surfaces work together to manipulate light, particularly by separating it into different colors—a phenomenon called dispersion.