Properties of Electrical Materials

Define conductivity and resistivity.

Conductivity \sigma measures how easily a material conducts current: J = \sigma E.
Resistivity \rho = 1/\sigma measures opposition to current flow. Units: \Omega \cdot m. [HB]

A copper wire has \rho = 1.68\times10^{-8}\ \Omega\cdot m and length 2 m, area 1\text{ mm}^2. Find resistance.

R = \rho\frac{L}{A} = (1.68\times10^{-8})\frac{2}{1\times10^{-6}} = 0.0336\ \Omega. [HB]

Explain how resistivity varies with temperature in conductors and semiconductors.

For metals: \rho_T = \rho_0[1+\alpha(T-T_0)] — resistivity increases with temperature.
For semiconductors: resistivity decreases as more charge carriers are thermally generated. [HB]

Define permittivity.

Permittivity \varepsilon measures how much electric field is reduced in a dielectric.

D = \varepsilon E and \varepsilon = \varepsilon_r \varepsilon_0. [HB]

What is dielectric strength?

The maximum electric field a material can withstand before breakdown, typically in kV/mm.

Define magnetic permeability.

Permeability \mu relates magnetic flux density and field intensity:

B = \mu H.

\mu = \mu_0 \mu_r, where \mu_r is relative permeability. [HB]

What is magnetic susceptibility \chi_m?

A measure of how easily a material becomes magnetized: M = \chi_m H.

For linear materials, B = \mu_0(1+\chi_m)H. [HB]

Describe ferromagnetic, paramagnetic, and diamagnetic materials.

Ferromagnetic: strong positive \mu_r (>1), e.g., iron.

Paramagnetic: weak \mu_r slightly >1.

Diamagnetic: \mu_r < 1; weakly repelled by magnetic fields.

What is hysteresis in magnetic materials?

Lag between B and H during magnetization cycles due to domain alignment. The loop area represents energy loss per cycle.

What is magnetic flux density?

B = \frac{\Phi}{A} — magnetic flux per unit area. Units: Tesla (T). [HB]

Define saturation magnetization.

The maximum magnetization a material can achieve when all domains are aligned.

What is eddy current loss?

Loss due to circulating currents in conductors exposed to changing magnetic fields; reduced by using laminated or powdered cores.

Define the Curie temperature.

The temperature above which ferromagnetic materials lose their magnetism and become paramagnetic.

What is the intrinsic carrier concentration of a semiconductor?

n_i = \sqrt{N_c N_v} e^{-E_g / 2kT} [HB] — increases exponentially with temperature.

Explain n-type and p-type doping.

n-type: donor atoms (Group V) add free electrons (majority carriers).
p-type: acceptor atoms (Group III) create holes (majority carriers).

At equilibrium in a p-n junction, what causes the depletion region?

Diffusion of carriers across the junction leaves behind charged ions, forming an electric field that opposes further diffusion.

Describe forward and reverse bias in a diode.

Forward bias: reduces barrier, allows current.
Reverse bias: increases barrier, minimal current until breakdown.

Define drift current.

Current due to movement of carriers in response to an electric field: J_{drift} = qn\mu_nE for electrons, J_p = qp\mu_pE for holes. [HB]

Define diffusion current.

Current due to carrier concentration gradients:

J_{diff,n} = qD_n\frac{dn}{dx}, \quad J_{diff,p} = -qD_p\frac{dp}{dx}. [HB]

State the Einstein relation.

\frac{D}{\mu} = \frac{kT}{q} — connects diffusion coefficient and mobility. [HB]

Find current density in n-type Si: n = 5\times10^{21}\ \text{m}^{-3},\ \mu_n = 0.135\ \text{m}^2/\text{V·s},\ E = 100\ \text{V/m}.

J = qn\mu_nE = (1.6\times10^{-19})(5\times10^{21})(0.135)(100)=108\ \text{A/m}^2. [HB]

What is carrier mobility?

Proportionality between drift velocity and electric field: v_d = \mu E.
Higher mobility ⇒ higher conductivity. [HB]

Define conductivity of a semiconductor.

\sigma = q(n\mu_n + p\mu_p) — sum of contributions from electrons and holes. [HB]

Explain what the Fermi level represents.

The energy at which the probability of electron occupancy is 50%.
Determines electron and hole concentrations in semiconductors.

What happens to the Fermi level in n-type vs p-type materials?

n-type: E_f shifts toward conduction band.

p-type: E_f shifts toward valence band.

What is a depletion region capacitance in a p-n junction?

C_j = \frac{\varepsilon A}{W}, where W is depletion width. [HB]

Define tunneling in semiconductors.

Quantum mechanical effect where carriers pass through a potential barrier, dominant in heavily doped (n++–p++) junctions.

State the ideal diode equation.

I = I_s(e^{qV/kT} - 1) [HB]

Explain what avalanche breakdown is.

Occurs when reverse-bias voltage causes carrier multiplication due to impact ionization.

What is Zener breakdown?

Occurs at low voltages due to quantum tunneling through a thin depletion region.

Define thermal conductivity.

Q = -kA\frac{dT}{dx} — rate of heat conduction through a material. [HB]

Compare thermal conductivities of metals, ceramics, and polymers.

Metals: high (free electrons).
Ceramics: moderate.
Polymers: low due to amorphous molecular structure.

Define coefficient of thermal expansion (CTE).

\alpha = \frac{1}{L}\frac{dL}{dT} — fractional change in length per temperature unit. [HB]

A 2 m aluminum rod (\alpha = 23\times10^{-6}/^\circ C) heats from 20°C to 120°C. Find expansion.

\Delta L = \alpha L \Delta T = 23\times10^{-6}(2)(100) = 0.0046\ \text{m}.
Expands 4.6 mm. [HB]

What is thermal runaway?

Positive feedback where increased temperature raises current, which further raises temperature — can cause device failure.

Define electrical noise.

Random fluctuations in current or voltage, e.g., Johnson (thermal) noise, shot noise, flicker noise.

State the formula for thermal (Johnson) noise voltage.

V_n = \sqrt{4kTRB} — depends on temperature, resistance, and bandwidth. [HB]

What is shot noise?

Noise due to discrete charge carriers crossing a barrier, proportional to DC current: i_n = \sqrt{2qIB}. [HB]

Describe superconductivity.

Phenomenon where resistivity drops to zero below a critical temperature T_c.
Magnetic fields are expelled (Meissner effect).

Explain why copper is used for electrical wiring.

Low resistivity, high conductivity, good ductility, and corrosion resistance.

Why is aluminum used in transmission lines?

Lower density → lighter weight, acceptable conductivity, lower cost than copper.

What is the approximate band gap of silicon at room temperature?

E_g = 1.12\text{ eV} — determines intrinsic carrier concentration.

Define piezoelectric effect.

Generation of electric potential when mechanical stress is applied to certain crystals (e.g., quartz, PZT).

Define thermoelectric effect.

Conversion between temperature difference and electric voltage (Seebeck and Peltier effects).

Explain the difference between intrinsic and extrinsic semiconductors.

Intrinsic: pure, equal electron and hole concentrations.
Extrinsic: doped with impurities to increase carrier concentration.

If the resistivity of copper doubles after heating, what happens to its conductivity?

Since \sigma = 1/\rho, conductivity is halved.
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