Refraction through plane surfaces (basic ideas)
Refraction through Plane Surfaces (Basic Ideas)
Refraction is the bending of light as it passes from one transparent medium to another. This phenomenon occurs due to the change in speed of light in different media. When light travels from a medium of a given refractive index to a medium with a different refractive index, its speed and direction change, except when it hits the boundary at a 90-degree angle (normal incidence).
Basic Principles of Refraction
- Snell's Law: This law gives the relationship between the angles of incidence and refraction when referring to light or other waves passing through a boundary between two different isotropic media.
[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) ]
where ( n_1 ) and ( n_2 ) are the refractive indices of the first and second medium, respectively, and ( \theta_1 ) and ( \theta_2 ) are the angles of incidence and refraction, respectively.
- Refractive Index: The refractive index of a medium is a measure of how much the speed of light (or other waves) is reduced inside the medium compared to the speed of light in a vacuum.
[ n = \frac{c}{v} ]
where ( c ) is the speed of light in a vacuum and ( v ) is the speed of light in the medium.
- Light Speed: Light travels faster in a less dense medium and slower in a denser medium. When light enters a denser medium, it bends towards the normal, and when it enters a less dense medium, it bends away from the normal.
Refraction at Plane Surfaces
When light rays pass through a plane surface (a flat boundary between two media), the following occurs:
- The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.
- The angle of incidence is related to the angle of refraction according to Snell's Law.
Formulas for Refraction through Plane Surfaces
- Lateral Shift: When a ray of light passes through a plane slab of thickness ( t ) and refractive index ( n ), it undergoes a lateral shift ( d ) which can be calculated using the formula:
[ d = t \cdot \frac{\sin(\theta_i - \theta_r)}{\cos(\theta_r)} ]
where ( \theta_i ) is the angle of incidence and ( \theta_r ) is the angle of refraction.
- Apparent Depth: If an object is viewed vertically from above, it appears to be raised (or shallower) due to refraction. The apparent depth ( d' ) can be calculated using:
[ d' = \frac{d}{n} ]
where ( d ) is the real depth and ( n ) is the refractive index of the medium in which the object is placed.
Differences and Important Points
| Property | Refraction at Plane Surfaces |
|---|---|
| Angles | The angle of incidence (( \theta_i )) and the angle of refraction (( \theta_r )) are related by Snell's Law. |
| Path | The path of the light ray is bent at the interface, except for normal incidence. |
| Speed | The speed of light changes as it enters a medium with a different refractive index. |
| Direction | Light bends towards the normal when entering a denser medium and away from the normal when entering a less dense medium. |
| Lateral Shift | A ray of light undergoes a lateral shift when passing through a slab due to the change in speed and direction. |
| Apparent Depth | Objects appear shallower when viewed from above in a denser medium due to refraction. |
Examples
Example 1: Calculating the Angle of Refraction
A light ray passes from air (refractive index = 1) into water (refractive index = 1.33) with an angle of incidence of 30 degrees. Find the angle of refraction.
Using Snell's Law:
[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) ]
[ 1 \cdot \sin(30^\circ) = 1.33 \cdot \sin(\theta_2) ]
[ \sin(\theta_2) = \frac{\sin(30^\circ)}{1.33} ]
[ \theta_2 = \arcsin\left(\frac{0.5}{1.33}\right) \approx 22.09^\circ ]
Example 2: Determining Lateral Shift
A light ray passes through a glass slab with a thickness of 10 cm and a refractive index of 1.5. If the angle of incidence is 45 degrees and the angle of refraction is 28.07 degrees, calculate the lateral shift.
[ d = t \cdot \frac{\sin(\theta_i - \theta_r)}{\cos(\theta_r)} ]
[ d = 10 \cdot \frac{\sin(45^\circ - 28.07^\circ)}{\cos(28.07^\circ)} ]
[ d \approx 10 \cdot \frac{\sin(16.93^\circ)}{\cos(28.07^\circ)} ]
[ d \approx 10 \cdot \frac{0.2924}{0.8829} \approx 3.31 \text{ cm} ]
Understanding refraction through plane surfaces is crucial for explaining many optical phenomena and is fundamental in the design of lenses, prisms, and various optical devices.