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X-Y plane is the boundary separating two transparent media with U1 and U2 .incident ray is A=6√3i+8√3j-10k.find unit vector along refracted ray?

X-Y plane is the boundary separating two transparent media with U1 and U2 .incident ray is A=6√3i+8√3j-10k.find unit vector along refracted ray?

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Grade:12th pass

1 Answers

Arun
25763 Points
3 years ago
Dear Amit
 
The incident angle is the angle the vector makes with the z-axis (that is the direction of the normal to the x-y plane). That angle θzi = arccos(k-component / |A|) 

|A| = √[(6√3)² + (8√3)² + 10²)] = √[3*36 + 3*64 + 100] = 20 

θzi = arccos(-10/20) = 60º 

The refracted angle will be sin(θzr)/sin(θzi) = n1/n2 = √2/√3 

sin(θzr) = √2/√3 * sin60º = √2/√3 * √3/2 = √2/2 

θzr = 45º 

cos45º = k-comp / |A| 

for a unit vector, |A| = 1 

k-comp = cos45º = √2/2 

The other two components must be adjusted by a factor r to result in a unity magnitude: 

√[r²*(3*36 + 3*64) + 0.5] = 1 

300*r² = 0.5 

r = 1/√600 

The new vector is then 

6/√200 i + 8/√200 j - √2/2 k 
 
Regards
Arun (askIITians forum expert)

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