${\displaystyle {\frac {\lambda _{2}}{\lambda _{1}}}={\frac {n_{1}}{n_{2}}}}$

${\displaystyle c=\lambda f}$

${\displaystyle \lambda =c/f}$

${\displaystyle \lambda _{2}n_{2}=\lambda _{1}n_{1}}$

${\displaystyle c/f_{2}n_{2}=c/f_{1}n_{2}}$

${\displaystyle {\frac {c}{f_{2}}}n_{2}={\frac {c}{f_{1}}}n_{1}}$

A1. Laser light of frequency 1.0 10 15 Hz is fired into a bucket of water. (Inside the water, the laser

has the same frequency as outside.)

A) If the index of refraction of the water is 1.33, find the wavelength of the laser trasmitted *in*

the water. (Warning: this will be different from the wavelength outside the water; think

carefully about what is different inside the water.)

${\displaystyle f_{1}=1.0\cdot 10^{15}}$

${\displaystyle n_{1}=1.000293}$

${\displaystyle c\longmapsto \lambda f}$

${\displaystyle \lambda }$ is the wavelength and ${\displaystyle f}$ is the frequency in hertz or inverse seconds and

${\displaystyle 299,792,458{\frac {m}{s}}=(2.2542.2547\times 10^{}-7m)\cdot }$

${\displaystyle {\frac {\sin \theta _{2}}{\sin \theta _{1}}}={\frac {v_{2}}{v_{1}}}={\frac {n_{1}}{n_{2}}}}$

The answer to A1 is ${\displaystyle (2.2542.2547\times 10^{-7}m)}$ and

${\displaystyle E=hf={\frac {hc}{\lambda }}}$

${\displaystyle {}_{5}^{11}\!B+{}_{1}^{1}\!proton={}_{6}^{12}\!totalmass=3\times {}_{2}^{4}\!He}$

${\displaystyle {}_{proton\#}^{total\#ofnucleons}\!nucleons}$

The total number of protons and neutrons in Boron-11 is 11. The total number of protons in boron is 5 because boron's atomic number ${\displaystyle Z}$ is 5. The total number of protons in boron-11 and a proton is 12 (11+1).

${\displaystyle {\frac {\lambda _{2}}{\lambda _{1}}}={\frac {n_{1}}{n_{2}}}={\frac {f_{1}}{f_{2}}}={\frac {\sin(\theta _{2})}{\sin(\theta _{1})}}={\frac {v_{2}}{v_{1}}}}$

Quantity	Symbol	Unit	Unit Symbol
Wavelength	${\displaystyle \lambda }$	meters	m
Index of refraction	${\displaystyle n}$	dimensionless
frequency	${\displaystyle f}$	hertz	Hz
angle	${\displaystyle \theta }$	radians	rad
velocity in medium	${\displaystyle v}$	meters per second	m/s

${\displaystyle \sin(\theta _{1})n_{1}=\sin(\theta _{2})n_{2}}$

The

${\displaystyle n1=n2=1.000293}$

${\displaystyle \sin(\theta _{1})=\sin(\theta _{2})}$

${\displaystyle \theta _{1}=\theta _{2}}$

${\displaystyle \theta _{1}+\theta _{2}+90^{\circ }=180^{\circ }}$

The two equations are ${\displaystyle \sin(\theta _{1})n_{1}=\sin(\theta _{2})n_{2}}$, which is simplified to ${\displaystyle \theta _{1}=\theta _{2}}$ and ${\displaystyle (\theta _{1}+\theta _{2})+90^{\circ }=180^{\circ }}$

${\displaystyle (\theta _{1}+\theta _{2})+90^{\circ }=180^{\circ }}$is subtracted from ${\displaystyle \theta _{1}+\theta _{2}=0}$ . we now have ${\displaystyle 2(\theta _{1}+\theta _{2})}$