how do i solve for n2?
1/ λ= 1.097x10^7m^-1 (1/(3)^2- 1/(n2)^2)
To solve for n2, we first need to rearrange the equation and isolate the term containing n2. Let's break down the steps:
1. Start with the given equation: 1/λ = 1.097x10^7 m^-1 * (1/(3)^2 - 1/(n2)^2)
2. Simplify the expression inside the parentheses: (1/(3)^2 - 1/(n2)^2)
3. Square the denominators: (1/9 - 1/n2^2)
4. Find a common denominator: (n2^2/9 - 1/n2^2)
5. Combine the fractions over the common denominator: (n2^2 - 9) / (9 * n2^2)
6. Substitute the values back into the original equation: 1/λ = 1.097x10^7 m^-1 * (n2^2 - 9) / (9 * n2^2)
7. Cross multiply to eliminate fractions: 1/λ * 9 * n2^2 = 1.097x10^7 m^-1 * (n2^2 - 9)
8. Distribute the terms on the right side of the equation: (9 * n2^2 / λ) = 1.097x10^7 m^-1 * n2^2 - 9 * 1.097x10^7 m^-1
9. Simplify the right side: (9 * n2^2 / λ) = (1.097x10^7 m^-1 * n2^2) - (9 * 1.097x10^7 m^-1)
10. Group the n2^2 terms on one side: (9 * n2^2 / λ) - (1.097x10^7 m^-1 * n2^2) = - (9 * 1.097x10^7 m^-1)
11. Factor out n2^2: n2^2 * [9/λ - 1.097x10^7 m^-1] = - (9 * 1.097x10^7 m^-1)
12. Divide both sides by the coefficient of n2^2: n2^2 = - (9 * 1.097x10^7 m^-1) / [9/λ - 1.097x10^7 m^-1]
13. Simplify the right side further if possible. Since the value of λ is not provided in your question, you may substitute a specific value for λ to get a numerical answer.
To solve for n2 in the equation 1/λ = 1.097x10^7m^-1 (1/(3)^2 - 1/(n2)^2), follow these steps:
Step 1: Simplify the equation:
1/λ = 1.097x10^7m^-1 (1/9 - 1/(n2)^2)
Step 2: Multiply both sides of the equation by λ to isolate the right side:
1 = 1.097x10^7m^-1 (1/9 - 1/(n2)^2)λ
Step 3: Divide both sides of the equation by 1.097x10^7m^-1:
1 / 1.097x10^7m^-1 = 1/9 - 1/(n2)^2)λ
Step 4: Simplify the left side of the equation:
9.11x10^-8m = 1/9 - 1/(n2)^2)λ
Step 5: Rearrange the equation to solve for (n2)^2:
1/(n2)^2 = 1/9 - 9.11x10^-8m/λ
Step 6: Find the reciprocal of both sides:
(n2)^2 = 1 / (1/9 - 9.11x10^-8m/λ)
Step 7: Simplify the right side of the equation:
(n2)^2 = 1 / (1/9 - 9.11x10^-8m/λ)
Step 8: Invert the fraction on the right side by multiplying the numerator and denominator by 9:
(n2)^2 = 9 / (9/9 - 9(9.11x10^-8m)/λ)
Step 9: Simplify the right side of the equation:
(n2)^2 = 9 / (1 - 9(9.11x10^-8m)/λ)
Step 10: Take the square root of both sides to solve for n2:
n2 = √(9 / (1 - 9(9.11x10^-8m)/λ))
Therefore, the solution for n2 is given by this equation.