Differentiation?

Question

Malus' Law for two polarizers in a row is:

I = Io cos^2 (angle)cos^2 (90degrees- angle).

Remembering that cos (90degrees- angle)= sin (angle), and

sin^2(angle)= 1-cos^2 (angle) find out where I is a maximum by differentiating I with respect to angle and setting

dI/d(angle)=0, and solving for angle.

Thanks for your help!

Answer 1

You really need to know calculus to do physics.

so I =lo cos^2 a sin^2 a  = lo/4 (sin 2a )^2

dI/da = lo/4 (4 sin 2a cos 2a)   = lo/2 sin 4a

sin 4a is zero when a=0, a=pi/8, ....

hope I haven't made a mistake ...

Answer 2

I = I0 cos^2(q)*cos^2(90 - q)

Take derivatie and set equal to zero:

dI/df = 0 = 2sin(q)cos(q) cos^2(90-q) - cos^2(q)2sin(90-q)cos(90-q)

0 = sin(q) cos(90-q) - cos(q)sin(90-q)

0 = sin(q)sin(q) - cos(q)cos(q) --> sin(q) = cos(q)

or tan(q) = 1  ---> q = pi/4, 7pi/4, ... or q = 90, 315, ...