Solving for Interference Minima in N-Slit Electric Field: A Step-by-Step Guide

[shaiqbashir]

Hi!

please help me in the following problem?

"The electric field at point P due to N slits is,

E (r, theta, t) = A(r) sin (N delta phi/2) cos(kr-wt)
_____________
sin (delta phi/2)

where delta phi=kdsin (theta) , "d" being the spacing between the adjacent slits. let N=4

(a) Find two angles on either side of theta=0 at which there is an interference minima.
(b) Find the spacing between the minima compare to the case of only two slits of separation "d".


Now i have solve its (a) part but i just can't solve its (b) part.

Can u please help me in this pleasezzzzzzzzzzzzzzzz.

Thanks in advance!

 

[stunner5000pt]

please post this in the homework help part of the forum

 

[R0man]

Hi there,

Sure, I'd be happy to help you with this problem. Solving for interference minima in N-slit electric field can be a bit tricky, but with a step-by-step guide, it should become clearer. Here's how you can approach this problem:

Step 1: Understand the given equation
The given equation represents the electric field at point P due to N slits. The variables A, r, theta, t, and k have specific meanings and units. Make sure you understand what each variable represents before proceeding.

Step 2: Simplify the equation
To make it easier to work with, you can simplify the equation by dividing both the numerator and denominator by sin (delta phi/2). This will give you a simpler equation:

E (r, theta, t) = A(r) cos(kr-wt)

Step 3: Set N = 4
Since the question asks for N = 4, you can substitute this value in the simplified equation:

E (r, theta, t) = A(r) cos(2 delta phi)

Step 4: Find the angles for interference minima
To find the angles for interference minima, you need to set the electric field to zero. This means that cos(2 delta phi) = 0. We know that cos(2x) = 0 at x = pi/4, 3pi/4, 5pi/4, etc. So, in this case, 2 delta phi = pi/4, 3pi/4, 5pi/4, etc. Substituting delta phi = kdsin(theta), we get:

2kdsin(theta) = pi/4, 3pi/4, 5pi/4, etc.

Solving for theta, we get:

theta = arcsin (pi/8kd), arcsin (3pi/8kd), arcsin (5pi/8kd), etc.

Step 5: Find the spacing between the minima
To find the spacing between the minima, we need to compare it to the case of only two slits of separation "d". In the case of two slits, the minima occur at theta = arcsin (kd), 3arcsin (kd), 5arcsin (kd), etc. So, the spacing between the minima is:

arcsin (kd), 2ar