Diffraction grating angular separation

[charlesh]

Homework Statement

Light consisting of two nearly equal wavelengths L and L+dL, where dL << L, is incident on a diffraction grating. The slit separation of the grating is D. Show that the angular separation of these two wavelengths in the m'th order is dTheta = dL / ((D/m)²+L²)^0.5

Homework Equations

D sin(Theta) = mL (m=0,+-1, +-2,...)

The Attempt at a Solution

D sin(Theta1)=m(L+dL)
D sin(Theta2)=m(L)
dTheta = sin(Theta1)-sin(Theta2) = (m/D)(L+dL-L) = (m/D)dL= dL/(D/m)
This answer doesn't match the formula I need to proof: dTheta = dL / ((D/m)²+L²)^0.5

 

[wukunlin]

Homework Statement

Light consisting of two nearly equal wavelengths L and L+dL, where dL << L, is incident on a diffraction grating. The slit separation of the grating is D. Show that the angular separation of these two wavelengths in the m'th order is dTheta = dL / ((D/m)²+L²)^0.5

Homework Equations

D sin(Theta) = mL (m=0,+-1, +-2,...)

The Attempt at a Solution

D sin(Theta1)=m(L+dL)
D sin(Theta2)=m(L)
dTheta = sin(Theta1)-sin(Theta2) = (m/D)(L+dL-L) = (m/D)dL= dL/(D/m)
This answer doesn't match the formula I need to proof: dTheta = dL / ((D/m)²+L²)^0.5

highlighted part is incorrect

 

[charlesh]

Thanks for the response.
For small angle, sin(Theta) is very close to Theta (in radian).