Question

In Young's double-slit experiment with a distance between the slits d and the angle between the principal axis, what is exp(ja) equal to, where j is the square root of unity?

a. cos ja+j sin a
b. sin a+j cos a
c. cos j⋅a+sin j
d. cos a+j sin a

Answer 1

In Young's double-slit experiment, exp(ja) corresponds to cos(a) + j sin(a) based on Euler's formula, which is essential in analyzing wave interference patterns using phases of waves. Constructive interference occurs when d sin θ = mλ, allowing prediction of interference fringe positions on a screen.

Step-by-step explanation:

In Young's double-slit experiment, the expression exp(ja) where j is the imaginary unit (often denoted as i in mathematics), represents Euler's formula, which relates complex exponentials to trigonometric functions. In this context, the correct form of exp(ja) translates to cos(a) + j sin(a). This equation is fundamental in physics when dealing with wave interference patterns as it allows the use of complex numbers to represent phases of waves.

The distance d between the slits plays a critical role in constructing or destructing interference patterns. According to the experiment, constructive interference occurs when d sin θ = mλ, where θ is the angle relative to the incident direction, m is the order of the interference (an integer), and λ is the wavelength of light used. The equation can be used to predict the position of interference fringes on a screen placed a distance x away from the slits.

When analyzing interference patterns, especially for small angles, one can approximate sin θ to θ itself (when in radians), making the mathematics simpler. This approximation is particularly useful in situations where d is much larger than the wavelength, leading to numerous evenly spaced bright spots on the screen.