Solving for Nodal Lines in a Two Point Source Interference Pattern

[lemin_rew]

Homework Statement

A two point source interference pattern is generated in a ripple tank by identical sources vibrating in phae and located 12.0cm apart. there are seven nodal lines observed on each side of the centre line. if the wavelength of the sources is doubled and they remain in phase

a) the number of nodal lines observed doubles
b) the frequency doubles
c) the speed of the wave doubles
d) the number of nodal lines decreases to half
e) the average distance between nodal lines decreases

Homework Equations

sintheta = Xn/L = (n-1/2) λ/d

am i solving for n? or Xn

The Attempt at a Solution

im not sure how to solve this mathematically.
the answer is d)
please help me

 

[anathema]

!

Hello,

Thank you for your question.

To solve this problem, we need to use the equation you provided: sintheta = Xn/L = (n-1/2) λ/d. In this equation, we are solving for n, which represents the number of nodal lines observed.

To start, let's review what we know from the problem:
- We have two identical sources vibrating in phase
- The sources are located 12.0cm apart
- There are seven nodal lines observed on each side of the centre line

Using the equation, we can rearrange it to solve for n: n = (2Lsintheta/d) + 1/2.

Since we are doubling the wavelength, we can substitute 2λ for λ in the equation. This gives us: n = (2Lsintheta/d) + 1/2.

Now, we can plug in the values we know:
- L = 12.0cm
- sintheta = 0 (since we are looking at the centre line)
- d = 12.0cm

This gives us: n = (2(12.0cm)(0)/12.0cm) + 1/2 = 1/2.

Therefore, when we double the wavelength, the number of nodal lines decreases to half, which is option d).

I hope this helps. Let me know if you have any further questions.