Finding the minimum angle of deviation for an equilateral prism

[thezac11]

Homework Statement

Use the formula: n=(sin(A+D)/2)/(sin(A)/2) to find D for an equilateral prism of index of refraction n=1.50

(where A=apex angle for a prism=60 degrees in this case, and D=minimum angle of deviation)

Homework Equations

Snell's Law: (n1)sin(i)=(n2)sin(r) , but i don't think this equation is needed for this problem.

The Attempt at a Solution

This is what I've got, but I know it is not completely correct. Any help would be greatly appreciated:

1.5=(sin(60+D)/2)/(sin(60)/2)

----> cancel the 2's and imput sin(60)cos(D)+cos(60)sin(D) for sin(60+D)

1.5=(sin(60)cos(D)+cos(60)sin(D))/(sin(60))

----> multiply numerator and denominator by cos(D)

1.5=(sin(60)cos(D)+cos(60)sin(D)cos(D))/(sin(60)cos(D))

----> cancel sin(60)cos(D) from numerator and denominator

1.5=cos(60)sin(D)cos(D)

1.5=(0.5)sin(D)cos(D)

3=sin(D)cos(D)

3=(0.5)(2sin(D)cos(D))

3=(0.5)sin(2D)

6=sin(2D)

?

 

[supratim1]

hey!

you cannot cancel the 2! in your second step.

its the angle (A+D)/2 and A/2 whose sine we are taking.

 

[thezac11]

so i didn't cancel the 2's and i came up with:

1.5=((cos(60)sin(D)cos(D))/2)/(1/2)

1.5=((1/2sin(D)cos(D))/2)x2

0.75=(1/2sin(D)cos(D))/2

1.5=(1/2sin(D)cos(D))

1.5=1/2(1/2(2sin(D)cos(D)))

3=1/2(1/2sin(2D))

6=1/2sin(2D)

12=sin(2D)

?

 

[supratim1]

you again did a mistake. consider angle A/2 = x and D/2 as y, and then proceed. you will grasp the idea.

 

[rwisz]

I would like to provide some feedback on the solution attempted above. Firstly, the use of Snell's Law is not necessary in this case as the problem only involves finding the minimum angle of deviation for an equilateral prism. The formula provided in the homework statement is sufficient for solving this problem.

Secondly, it is important to note that the formula provided is in terms of the index of refraction, not the angle of incidence or refraction. Therefore, the equation should be rearranged to solve for D, the minimum angle of deviation. The correct equation should be:

D = 2sin^-1(nsin(A/2))

Where n is the index of refraction and A is the apex angle of the prism.

Finally, the solution attempted above is on the right track, but there are some errors in the calculations. The equation should be rearranged to isolate D, and then the inverse sine function should be applied to both sides to solve for D. The final equation should be:

D = sin^-1(3/n) - 30 degrees

Where n is the index of refraction. This will give the minimum angle of deviation in degrees.

In conclusion, to find the minimum angle of deviation for an equilateral prism, the formula D = sin^-1(3/n) - 30 degrees can be used, where n is the index of refraction. It is important to rearrange the equation properly and use the correct units (degrees or radians) for the inverse sine function.