Einstein Coefficients - Laser Beams

[Ivor Denham]

Homework Statement

Assume a two level laser system with no degeneracy (g1 = g2 = 1).
If [tex]N_2 + N1 = \text{constant}[\tex], show that [tex]B12 = B21[\tex].<br /> <br /> <h2>Homework Equations</h2><br /> [tex]\frac{\partial N_1}{\partial t}=-B_{12}\rho(v_{21}N_1[\tex]<br /> [tex]\frac{\partial N_2}{\partial t}=-B_{21}\rho(v_{21}N_2[\tex]<br /> <br /> <h2>The Attempt at a Solution</h2> <br /> I think there might be something up with the signs in the relevant equation, a convention I am not observing.<br /> <br /> [tex]N_1+N_2=c[\tex]<br /> <b>$$\frac{\partial N_1}{\partial t}+\frac{\partial N_2}{\partial t}=\frac{\partial c}{\partial t}[\tex]$$</b>[tex] <b><b><b>$$\frac{\partial N_1}{\partial t}+\frac{\partial N_2}{\partial t}=0[\tex]$$</b>[tex][/tex]</b>[tex] <b><b><b><b><b><b>$$-B_{12}\rho(v_{21}N_1+-B_{21}\rho(v_{21}N_2=0[\tex]$$</b>[tex][/tex]</b>[tex][/tex]</b>[tex][/tex]</b>[tex] <b><b><b><b><b><b><b><b><b><b><b><b>$$-B_{12}N_1+-B_{21}N_2=0[\tex]$$</b>[tex][/tex]</b>[tex][/tex]</b>[tex][/tex]</b>[tex][/tex]</b>[tex][/tex]</b>[tex][/tex]</b>[tex][/tex]</b>$$<b><b><b><b><b><b><b><b><br /> From here I am stuck, where have I gone wrong?</b></b></b></b></b></b></b></b>$$</b>[tex][/tex]</b>[tex][/tex]</b>[tex][/tex]</b>[tex][/tex][/tex]</b>[tex][tex][/tex][/tex]</b>[tex][tex][/tex][/tex][/tex]</b>[tex][tex]$$[/B]$$[/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex]

 

[Ivor Denham]

Also could someone tell me how to fix the latex?