Physics

$\beta =\frac{\lambda D}{d}=\frac{500*10^{-9}*1}{2*10^{-3}}=250*10^{-6}m=250*10^{-3}mm=0.25mm$

$N=mgcos30°+qEsin30°$

$a=\frac{mgsin\theta -qEcos\theta -\mu N}{m}=2.30m/s^2$

$S=ut+\frac{1}{2}a{t}^2$

$\Rightarrow t=\sqrt[]{\frac{2\mathcal{l}}{a}}=1.31sec$

From defined of bulk modulus

$K=-\frac{P}{\left(dV/V\right)}.......\left(1\right)$

m = $\rho v=constant$

$\rho dV+Vd\rho =0$

$\frac{dV}{V}=-\frac{d\rho }{\rho }.........\left(2\right)$

With the help of equations (1) and (2), we can write

$K=-\frac{P}{\left(-\frac{d\rho }{\rho }\right)}=\frac{\rho P}{d\rho }\Rightarrow d\rho =\frac{\rho P}{K}$

Frequency increases on filing. So, initial frequency of A is 335 Hz.

f = 340 – 5 = 335 Hz

$Z=\sqrt[]{R^2+{\left(X_L-X_C\right)}^2}$

$Z=\sqrt[]{{\left(120\right)}^2+{\left(10-100\right)}^2}=150\Omega$

$\omega =\frac{1}{\sqrt[]{LC}}=\frac{1}{\sqrt[]{10^{-1}*10^{-4}}}=\sqrt[]{10^5}$

$?\omega =2\pi f$

$\Rightarrow f=\frac{10^3}{2\pi \sqrt[]{10}}=50Hz$

$A=A_0{e}^{-\lambda t_1}$ [Radio active decay law]

$\frac{A}{5}=A_0{e}^{-\lambda \left(t_2-t_1\right)}$

$\Rightarrow ln5=\lambda \left(t_2-t_1\right)$

$\Rightarrow Averagelife=\frac{1}{\lambda }=\frac{\left(t_2-t_1\right)}{\mathcal{l}n5}$

Stress = Y * strain

=> $\frac{T_1}{A}=Y*-\frac{\left({\mathcal{l}}_1-\mathcal{l}\right)}{\mathcal{l}}\left(i\right)$

$\frac{T_2}{A}=Y*\frac{\left({\mathcal{l}}_2-\mathcal{l}\right)}{\mathcal{l}}\left(ii\right)$

=> $\frac{T_1}{T_2}=\frac{{\mathcal{l}}_1-\mathcal{l}}{{\mathcal{l}}_2-\mathcal{l}}$

$\Rightarrow \mathcal{l}=\frac{T_1{\mathcal{l}}_2-T_2{\mathcal{l}}_2}{T_1-T_2}=\frac{T_2{\mathcal{l}}_1-T_1{\mathcal{l}}_2}{T_2-T_1}$            

Lyman Series n_f = 1, ni = 2, 3.

Paschen series n_f = 3, ni = 4, 5, 6 -

$\frac{1}{{\lambda }_1}=R{Z}^2\left(\frac{1}{1^2}-\frac{1}{4^2}\right)$

$\frac{1}{{\lambda }_2}=R{Z}^2\left(\frac{1}{3^2}-\frac{1}{4^2}\right)$

$\frac{1/{\lambda }_1}{1/{\lambda }_2}=\frac{1-\frac{1}{16}}{\frac{1}{9}-\frac{1}{16}}\Rightarrow \frac{{\lambda }_2}{{\lambda }_1}=\frac{\frac{15}{16}}{\frac{7}{9*16}}=\frac{15*9}{7}\Rightarrow \frac{{\lambda }_1}{{\lambda }_2}=\frac{7}{135}$

$l=l_{1}sin\omega t+l_{2}cos\omega t=\sqrt[]{l_1^2+l_2^2}\left[\left(\frac{l_1}{\sqrt[]{l_1^2+l_2^2}}\right)sin\omega t+\frac{l_2}{\sqrt[]{l_1^2+l_2^2}}cos\omega t\right]$

$=\sqrt[]{l_1^2+l_2^2}sin\left(\omega t+\alpha \right)=l_{0}sin(\omega t+\alpha$

$l_{rms}=\frac{l_0}{\sqrt[]{2}}=\frac{\sqrt[]{l_1^2+l_2^2}}{\sqrt[]{2}}$