In Newton's ring experiment, the diameter of the 18th ring was found to be 0.59 cm and that of the 8th ring was 0.336 cm. If the radius of the plano-convex lens is 100 cm, calculat... In Newton's ring experiment, the diameter of the 18th ring was found to be 0.59 cm and that of the 8th ring was 0.336 cm. If the radius of the plano-convex lens is 100 cm, calculate the wavelength of the light used. Read more

Steps to Solve

1. Write down the given information

Diameter of the 18th ring, $D_{18} = 0.59 , \text{cm}$ Diameter of the 8th ring, $D_8 = 0.336 , \text{cm}$ Radius of curvature of the plano-convex lens, $R = 100 , \text{cm}$

2. Formula for the diameter of the $n^{th}$ ring

The formula for the diameter of the $n^{th}$ bright ring in Newton's rings experiment is given by: $$D_n^2 = 4nR\lambda$$ where $D_n$ is the diameter of the $n^{th}$ ring, $R$ is the radius of curvature of the lens, $\lambda$ is the wavelength of the light used, and $n$ is the ring number.

3. Apply the formula for the 18th and 8th rings

For the 18th ring ($n = 18$): $$D_{18}^2 = 4(18)R\lambda$$ $$(0.59 , \text{cm})^2 = 4(18)(100 , \text{cm})\lambda$$

For the 8th ring ($n = 8$): $$D_8^2 = 4(8)R\lambda$$ $$(0.336 , \text{cm})^2 = 4(8)(100 , \text{cm})\lambda$$

4. Subtract the two equations to eliminate unknowns

Subtract the equation for the 8th ring from the equation for the 18th ring: $$D_{18}^2 - D_8^2 = 4R\lambda(18 - 8)$$ $$(0.59)^2 - (0.336)^2 = 4(100)\lambda(10)$$ $$0.3481 - 0.112896 = 4000\lambda$$ $$0.235204 = 4000\lambda$$

5. Solve for $\lambda$

$$\lambda = \frac{0.235204}{4000}$$ $$\lambda = 0.000058801 , \text{cm}$$ $$\lambda = 5.8801 \times 10^{-5} , \text{cm}$$

6. Convert cm to Angstroms

Since $1 , \text{cm} = 10^8 , \text{Angstroms (Å)}$, $$\lambda = 5.8801 \times 10^{-5} \times 10^8 , \text{Å}$$ $$\lambda = 5880.1 , \text{Å}$$