Rearrange this expression for IXI0 to get an equation that g

Rearrange this expression for I(X)/I(0) to get an equation that gives 2X_1/2? Hint: Substitute X in the equation with X_1/2 and go from there.

Solution

I(x)/I(0) = exp[ -2k( R - sqrt(R^2 - x^2) ]

Given: I(x)/I(0) = 0.5

So, 0.5 =exp[ -2k( R - sqrt(R^2 - x^2) ]

take natural log on both sides:

ln(0.5) = [ -2k( R - sqrt(R^2 - x^2) ]

-0.69 =-2k( R - sqrt(R^2 - x^2)

0.346 = k(R - sqrt(R^2 - x^2)

0.346/k = R - sqrt(R^2 - x^2

sqrt(R^2 - x^2) = R - 0.346/k

square both sides:

R^2 - x^2 = (R - 0.346/k)^2

R^2 - x^2= R^2 + (0.346/k)^2 - 0.69/k

0.69/k - (0.346/k)^2 = x^2

Plug x = X1/2

So, we getr: (X1/2)^2 = 0.69/k - (0.346/k)^2

take squrae root on both sides:

X1/2 = [ 0.69/k - (0.346/k)^2]^1/2

2X1/2 = 2sqrt[ 0.69/k - (0.346/k)^2]