Current Electricity flash cards

1. 1.Define electric current and give its SI unit.

   Electric current is the rate of flow of charge: $I = \dfrac{dq}{dt}$. SI unit is the ampere (A); $1\,\text{A} = 1\,\text{C/s}$. It is a scalar with a conventional direction (direction of positive charge flow).

   Hint: Charge per unit time.

2. 2.What is conventional current direction versus electron flow?

   Conventional current is taken in the direction positive charges would move, i.e. from high to low potential (outside the source). Electrons actually drift opposite to this. Both describe the same physical current.

   Hint: Positive-charge convention vs real electron motion.

3. 3.Define drift velocity and give the current relation.

   Drift velocity $v_d$ is the average velocity electrons acquire along the wire under an applied field. Current $I = neAv_d$, where $n$ is free-electron density, $e$ the electronic charge, $A$ the cross-section.

   Hint: $I = neAv_d$.

4. 4.Relate drift velocity to the applied electric field and relaxation time.

   $v_d = \dfrac{eE}{m}\tau$, where $\tau$ is the mean relaxation time between collisions, $E$ the field, $m$ the electron mass. Drift velocity is proportional to the field.

   Hint: Acceleration $eE/m$ acting for time $\tau$.

5. 5.Define mobility of a charge carrier and its SI unit.

   Mobility $\mu = \dfrac{v_d}{E} = \dfrac{e\tau}{m}$, the drift speed per unit field. SI unit is $\text{m}^2\,\text{V}^{-1}\,\text{s}^{-1}$.

   Hint: Drift speed per unit field.

6. 6.Express current density $J$ in terms of $n$, $v_d$ and in terms of conductivity.

   $J = \dfrac{I}{A} = nev_d$. Also $\vec{J} = \sigma \vec{E}$ (microscopic Ohm's law), where $\sigma$ is conductivity.

   Hint: $J = nev_d = \sigma E$.

7. 7.State Ohm's law in its macroscopic and microscopic forms.

   Macroscopic: $V = IR$. Microscopic: $\vec{J} = \sigma\vec{E}$. Ohm's law states current is proportional to potential difference at constant temperature.

   Hint: $V=IR$ and $J=\sigma E$.

8. 8.Define resistance and give the formula for a uniform conductor.

   Resistance $R = \dfrac{V}{I}$; SI unit ohm ($\Omega$). For a uniform wire $R = \dfrac{\rho L}{A}$, with $\rho$ resistivity, $L$ length, $A$ area.

   Hint: $R = \rho L / A$.

9. 9.Define resistivity and relate it to microscopic quantities.

   Resistivity $\rho = \dfrac{RA}{L}$; SI unit $\Omega\,\text{m}$. Microscopically $\rho = \dfrac{m}{ne^2\tau}$, independent of dimensions, dependent on material and temperature.

   Hint: $\rho = m/(ne^2\tau)$.

10. 10.How do resistance and resistivity change when a wire is stretched to $n$ times its length (volume constant)?

    Resistivity $\rho$ is unchanged (material property). Length becomes $nL$, area becomes $A/n$, so $R' = \rho\dfrac{nL}{A/n} = n^2 R$. Resistance increases $n^2$ times.

    Hint: $R \propto L^2$ at fixed volume.

11. 11.State the limitations/failures of Ohm's law.

    Ohm's law fails when: $V$ is non-linear in $I$ (e.g. diodes), $V$–$I$ relation depends on sign of $V$ (rectifiers), or is multivalued. Such devices are non-ohmic; also fails at high fields/temperatures.

    Hint: Non-linear, direction-dependent, or multivalued $V$–$I$.

12. 12.Give the temperature dependence of resistivity for a conductor.

    $\rho_T = \rho_0[1 + \alpha(T - T_0)]$, where $\alpha$ is the temperature coefficient of resistivity. For metals $\alpha > 0$ (resistivity rises with temperature).

    Hint: $\rho = \rho_0(1+\alpha\Delta T)$.

13. 13.How does resistivity of metals, semiconductors, and conductors differ with temperature?

    Metals: $\rho$ increases with $T$ ($\alpha>0$). Semiconductors and insulators: $\rho$ decreases with $T$ ($\alpha<0$) as more carriers are freed. Alloys like nichrome/manganin have very small $\alpha$.

    Hint: Metals up, semiconductors down.

14. 14.Why does resistivity of a metal increase with temperature (microscopic reason)?

    $\rho = \dfrac{m}{ne^2\tau}$. In metals $n$ is nearly constant, but higher temperature causes more frequent collisions, reducing relaxation time $\tau$, so $\rho$ rises.

    Hint: $\tau$ falls as $T$ rises.

15. 15.Define temperature coefficient of resistance $\alpha$.

    $\alpha = \dfrac{R_T - R_0}{R_0\,(T - T_0)} = \dfrac{1}{R}\dfrac{dR}{dT}$; the fractional change in resistance per degree rise in temperature. Unit: $\text{K}^{-1}$ or $^\circ\text{C}^{-1}$.

    Hint: Fractional resistance change per degree.

16. 16.Derive the formula for resistors in series.

    Same current $I$ flows through each; total voltage $V = V_1+V_2+\dots = I(R_1+R_2+\dots)$. Hence $R_s = R_1 + R_2 + R_3 + \dots$. Series resistance exceeds the largest resistor.

    Hint: Voltages add, current common.

17. 17.Derive the formula for resistors in parallel.

    Same voltage $V$ across each; currents add: $I = V\left(\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dots\right)$. So $\dfrac{1}{R_p} = \dfrac{1}{R_1}+\dfrac{1}{R_2}+\dots$. Parallel resistance is less than the smallest.

    Hint: Currents add, voltage common.

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