A parallel beam of light enters into a clear plastic bead 2.5 cm diameter and refractive index of 1.44. The point beyond the bead where the focus is

When a parallel beam of light enters a medium like a clear plastic bead, the behavior of the light is influenced by the refractive index of the material. In this case, we have a bead with a diameter of 2.5 cm and a refractive index of 1.44. To find the point beyond the bead where the light focuses, we can apply some principles of optics, particularly those related to lenses and refraction.

Understanding Refraction in the Bead

As the parallel beam of light enters the bead, it slows down due to the higher refractive index compared to air. This change in speed causes the light to bend, or refract, at the boundary between the air and the plastic. The degree of bending is determined by Snell's Law, which states:

n₁ * sin(θ₁) = n₂ * sin(θ₂)

Where:

• n₁ is the refractive index of the first medium (air, approximately 1.00).
• θ₁ is the angle of incidence.
• n₂ is the refractive index of the second medium (the plastic bead, 1.44).
• θ₂ is the angle of refraction.

Calculating the Focal Point

For a spherical bead, the light rays will converge to a point after exiting the bead. To find this focal point, we can use the lens maker's equation, which is applicable here since the bead acts like a lens:

1/f = (n - 1) * (1/R₁ - 1/R₂)

In our case:

• The radius of the bead, R, is half of the diameter: R = 2.5 cm / 2 = 1.25 cm.
• Since the bead is spherical, we can assume R₁ = 1.25 cm (the front surface) and R₂ = -1.25 cm (the back surface, which is considered negative in this context).
• The refractive index n = 1.44.

Plugging these values into the lens maker's equation gives us:

1/f = (1.44 - 1) * (1/1.25 - 1/(-1.25))

Calculating this step-by-step:

• 1/f = 0.44 * (0.8 + 0.8) = 0.44 * 1.6 = 0.704
• Thus, f = 1 / 0.704 ≈ 1.42 cm.

Finding the Focus Beyond the Bead

The focal point is located on the opposite side of the bead from where the light enters. Since the focal length we calculated is approximately 1.42 cm, this means that the light will converge to a point about 1.42 cm from the surface of the bead on the side opposite to where the light entered.

To summarize, when a parallel beam of light enters a clear plastic bead with a diameter of 2.5 cm and a refractive index of 1.44, it will focus at a point approximately 1.42 cm beyond the bead. This phenomenon illustrates the principles of refraction and how lenses work, even in simple spherical shapes like beads.