Modern PhysicsHigh Yield (2 Qs in JEE)

Semiconductors & Logic Gates Formulas

P-N junction diode, Zener breakdown diode, rectifiers, and truth tables of Boolean logic gates (AND, OR, NOT, NAND, NOR).

8 Formulas•462+ Questions in Bank

Core Formulas & Mathematical Identities

Intrinsic & Extrinsic Carrier Concentrations

n_e \cdot n_h = n_i^2 \quad (\text{Mass Action Law}), \quad \sigma = e(n_e \mu_e + n_h \mu_h)

Product of electron and hole numbers remains constant at thermal equilibrium.

Examiner Tip: In n-type, n_e ≈ N_d; in p-type, n_h ≈ N_a.

Half-Wave & Full-Wave Rectifier Efficiency

\eta_{\text{half}} = \frac{0.406 R_L}{r_f + R_L} \le 40.6\%, \quad \eta_{\text{full}} = \frac{0.812 R_L}{r_f + R_L} \le 81.2\%

AC to DC power conversion efficiency.

Examiner Tip: Output ripple frequency of full-wave rectifier is 2f_in; for half-wave it is f_in.

De Morgan's Laws & Logic Operations

\overline{A + B} = \bar{A} \cdot \bar{B}, \quad \overline{A \cdot B} = \bar{A} + \bar{B}

Fundamental duality theorems for simplifying Boolean digital circuits.

Examiner Tip: NAND: Y = (A · B)' (0 only when both inputs are 1). NOR: Y = (A + B)' (1 only when both inputs are 0).

Zener Diode as Voltage Regulator

V_{\text{out}} = V_Z, \quad I_s = I_Z + I_L, \quad R_s = \frac{V_{\text{in}} - V_Z}{I_Z + I_L}

Maintains constant terminal load voltage V_Z regardless of unregulated DC input voltage fluctuations.

Examiner Tip: Zener operates in breakdown region in reverse bias. Minimum input voltage must exceed V_Z.

Transistor Current & Voltage Gains

I_E = I_B + I_C, \quad \beta = \frac{I_C}{I_B} = \frac{\alpha}{1 - \alpha}, \quad A_V = \beta \frac{R_{\text{out}}}{R_{\text{in}}}, \quad A_P = \beta^2 \frac{R_{\text{out}}}{R_{\text{in}}}

Common-emitter amplifier relationships where α < 1 (typically 0.95 to 0.99) and β is large (typically 20 to 100).

Examiner Tip: In common emitter amplifier, output voltage has a 180° (π radians) phase shift relative to input signal.

Dynamic Resistance & Rectifier Ripple Factor

r_d = \frac{\Delta V}{\Delta I}, \quad r = \sqrt{\left(\frac{I_{\text{rms}}}{I_{\text{dc}}}\right)^2 - 1} \implies r_{\text{half}} = 1.21, \quad r_{\text{full}} = 0.482

Differential resistance of P-N junction and effectiveness of rectifier in delivering pure DC voltage.

Examiner Tip: A smaller ripple factor indicates a purer DC output with less harmonic fluctuation.

Mass Action Law & Semiconductor Conductivity

n_e \cdot n_h = n_i^2, \quad \sigma = e(n_e \mu_e + n_h \mu_h), \quad \sigma_{\text{n-type}} \approx e N_D \mu_e, \quad \sigma_{\text{p-type}} \approx e N_A \mu_h

Carrier concentration product in thermal equilibrium and electrical conductivity dictated by electron (μ_e) and hole (μ_h) mobilities.

Examiner Tip: Electron mobility is always higher than hole mobility (μ_e > μ_h) in semiconductors.

Optoelectronic Bandgap Wavelength (LED, Photodiode)

\lambda_{\text{threshold}} = \frac{hc}{E_g} \approx \frac{12400}{E_g \text{ (in eV)}} \,\text{\AA}

Cut-off optical wavelength for light absorption in solar cells/photodiodes and photon emission in LEDs.

Examiner Tip: Photodiodes are operated in reverse bias because fractional change in reverse saturation current upon illumination is substantially higher than in forward bias.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Semiconductors & Logic Gates

•Zener diode is ALWAYS operated in REVERSE BIAS under breakdown voltage for voltage regulation.
•NAND and NOR gates are UNIVERSAL logic gates.

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