Engineering Physics
JNTUK R23 Board

For coherent light fields $E_1(t) = a_1 \sin(\omega t)$ and $E_2(t) = a_2 \sin(\omega t + \phi)$, the resultant intensity is given by: $$I = |E_1 + E_2|^2 = a_1^2 + a_2^2 + 2a_1 a_2 \cos(\phi) = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos(\phi)$$. Maximum intensity occurs when $\cos(\phi) = 1 \implies \phi = 2n\pi$, while minimum occurs at $\cos(\phi) = -1 \implies \phi = (2n+1)\pi$.

Enclosed air wedge has a thickness boundary $t = r^2 / (2R)$ where $R$ is curvature radius. The path difference under normal incidence (including phase shift $\pi$ from reflections at optically denser glass is): $$\Delta = 2t + \frac{\lambda}{2} = \frac{r^2}{R} + \frac{\lambda}{2}$$. For destructive dark rings (minimum light reflection): $\Delta = (2n+1)\frac{\lambda}{2} \implies \frac{r_n^2}{R} = n\lambda \implies r_n^2 = n\lambda R$. Therefore, the diameter of the $n$-th dark ring is: $$D_n = 2r_n = \sqrt{4n\lambda R} \implies D_n^2 = 4n\lambda R$$.

Lattice geometry is governed by 3 lattice translation vectors $(a, b, c)$ and 3 internal axial angles $(\alpha, \beta, \gamma)$. Crystalline structures require exact periodic translation: $\vec{T} = u\vec{a} + v\vec{b} + w\vec{c}$.

Atomic packing fraction is $APF = N_{\text{eff}} \times \left(\frac{4}{3}\pi R^3\right) / a^3$.
• Simple Cubic (SC): $N_{\text{eff}} = 1$. Curvature parameter $a = 2R$. $$APF = \frac{\pi}{6} = 0.52 = 52.4\%$$.
• Body-Centered Cubic (BCC): $N_{\text{eff}} = 2$. Diagonal relation $\sqrt{3}a = 4R \implies a = 4R/\sqrt{3}$. $$APF = 2 \times \frac{4}{3}\pi R^3 \times \left(\frac{\sqrt{3}}{4R}\right)^3 = \frac{\sqrt{3}\pi}{8} = 0.68 = 68\%$$.
• Face-Centered Cubic (FCC): $N_{\text{eff}} = 4$. Diagonal relation $\sqrt{2}a = 4R \implies a = 4R/\sqrt{2}$. $$APF = 4 \times \frac{4}{3}\pi R^3 \times \left(\frac{\sqrt{2}}{4R}\right)^3 = \frac{\pi}{3\sqrt{2}} = 0.74 = 74\%$$.

Inside an electric polarization material, the microscopic local field experienced by a molecule within a spherical boundary is: $$E_{\text{local}} = E_0 + \frac{P}{3\epsilon_0}$$. Here, $E_0$ is external applied macroscopic field, and $\frac{P}{3\epsilon_0}$ is the cavity polarization field derived by integration of normal surface polarizations over spherical domains.

Linking macroscopic relative permittivity $\epsilon_r$ to molecular polarizability $\alpha$: $$P = N \alpha E_{\text{local}} = N \alpha \left( E + \frac{P}{3\epsilon_0} \right)$$. Given that $P = \epsilon_0 (\epsilon_r - 1) E$, substituting this into the local field formula and solving yields the Clausius-Mossotti relation: $$\frac{\epsilon_r - 1}{\epsilon_r + 2} = \frac{N \alpha}{3\epsilon_0}$$.

The wave field function $\Psi(\vec{r}, t)$ must be continuous, single-valued, and have finite first derivatives. Normalized conditions require: $$\int_{-\infty}^{\infty} |\Psi|^2 dV = 1$$ (certainty of particle existing in space). Born's interpretation: $|\Psi|^2$ represents spatial probability distribution.

Confinement potential: $V(x)=0$ inside $(0 < x < L)$, and $V(x)=\infty$ outside. Schrödinger boundary formula: $$\frac{d^2 \psi}{dx^2} + \frac{2mE}{\hbar^2}\psi = 0 \implies \psi(x) = A\sin(kx) + B\cos(kx)$$.
Boundary $\psi(0) = 0 \implies B=0$. Boundary $\psi(L) = 0 \implies \sin(kL) = 0 \implies kL = n\pi$. Since $E = \frac{\hbar^2 k^2}{2m}$, we substitute $k = \frac{n\pi}{L}$ to prove that internal energies are highly quantized: $$E_n = \frac{n^2 h^2}{8mL^2}$$. Normalized wavefunction state: $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$.

An electron traveling inside a periodic potential lattice $V(x) = V(x+a)$ is modeled via modulated Bloch waves: $$\psi(k, x) = u_k(x) e^{ikx}$$. Here, $e^{ikx}$ represents normal plane wave, and $u_k(x)$ is periodic amplitude: $u_k(x+a)=u_k(x)$.

Under applied magnetic fields $B_z$ and current densities $J_x$, perpendicular electric force balances magnetic force: $$q E_y = q v_d B_z \implies E_y = v_d B_z$$.
Since current density is $J_x = n q v_d \implies v_d = J_x / (nq)$, substituting back yields: $$E_y = \frac{J_x B_z}{nq}$$. The Hall Coefficient $R_H$ is: $$R_H = \frac{E_y}{J_x B_z} = \frac{1}{nq}$$. Evaluating $R_H$ sign and size determines material doping signature (n-type vs p-type) as well as carrier densities $n$.