SUMMARY

1. Current through a given area of a conductor is the net charge passing per unit time through the area.
2. To maintain a steady current, we must have a closed circuit in which an external agency moves electric charge from lower to higher potential energy. The work done per unit charge by the source in taking the charge from lower to higher potential energy (i.e., from one terminal of the source to the other) is called the electromotive force, or emf, of the source. Note that the emf is not a force; it is the voltage difference between the two terminals of a source in open circuit.
3. Ohm’s law: The electric current \( I \) flowing through a substance is proportional to the voltage \( V \) across its ends, i.e., \( V \propto I \) or \( V=RI \), where \( R \) is called the resistance of the substance. The unit of resistance is ohm: \( 1\Omega=1VA^{-1} \).
4. The resistance \( R \) of a conductor depends on its length \( l \) and cross-sectional area \( A \) through the relation, \[ R=\frac{\rho l}{A} \] where \( \rho \), called resistivity is a property of the material and depends on temperature and pressure.
5. Electrical resistivity of substances varies over a very wide range. Metals have low resistivity, in the range of \( 10^{-8}\Omega m\) to \( 10^{-6}\Omega m\). Insulators like glass and rubber have \( 10^{22} \) to \( 10^{24} \) times greater resistivity. Semiconductors like \( Si \) and \( Ge \) lie roughly in the middle range of resistivity on a logarithmic scale.
6. In most substances, the carriers of current are electrons; in some cases, for example, ionic crystals and electrolytic liquids, positive and negative ions carry the electric current.
7. Current density \( \mathbf{j} \) gives the amount of charge flowing per second per unit area normal to the flow, \[ \mathbf{j}=n q \boldsymbol{v}_{\mathrm{d}} \] where \( n \) is the number density (number per unit volume) of charge carriers each of charge \( q \), and \( \boldsymbol{v}_d \) is the drift velocity of the charge carriers. For electrons \( q = – e \). If \( \mathbf{j} \) is normal to a cross-sectional area \( \mathbf{A} \) and is constant over the area, the magnitude of the current \( I \) through the area is \( nev_d A \).
8. Using \( E = V/l \), \( I = nev_d A \), and Ohm’s law, one obtains \[ \frac{e E}{m}=\rho \frac{n e^{2}}{m} v_{d} \] The proportionality between the force \( eE \) on the electrons in a metal due to the external field \( E \) and the drift velocity \( v_d \) (not acceleration) can be understood, if we assume that the electrons suffer collisions with ions in the metal, which deflect them randomly. If such collisions occur on an average at a time interval \( \tau \), \[ v_{d}=a \tau=e E \tau / m \] where \( a \) is the acceleration of the electron. This gives \[ \rho=\frac{m}{n e^{2} \tau} \]
9. In the temperature range in which resistivity increases linearly with temperature, the temperature coefficient of resistivity \( \alpha \) is defined as the fractional increase in resistivity per unit increase in temperature.
10. Ohm’s law is obeyed by many substances, but it is not a fundamental law of nature. It fails if
    1. \( V \) depends on \( I \) non-linearly.
    2. the relation between \( V \) and \( I \) depends on the sign of \( v \) for the same absolute value of \( V \).
    3. The relation between \( V \) and \( I \) is non-unique.
    An example of (a) is when \( \rho \) increases with \( I \) (even if temperature is kept fixed). A rectifier combines features (a) and (b). \( GaAs \) shows the feature (c).
11. When a source of emf \( \varepsilon \) is connected to an external resistance \( R \), the voltage Vext across \( R \) is given by \[ V_{e x t}=I R=\frac{\varepsilon}{R+r} R \] where \( r \) is the internal resistance of the source.
12. 1. Total resistance \( R \) of \( n \) resistors connected in series is given by \[ R=R_1+R_2+R_3+...+R_n \]
    2. Total resistance \( R \) of \( n \) resistors connected in parallel is given by \[ \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\ldots \ldots+\frac{1}{R_{n}} \]
13. Kirchhoff's Rules-
    1. Junction Rule: At any junction of circuit elements, the sum of currents entering the junction must equal the sum of currents leaving it.
    2. Loop Rule: The algebraic sum of changes in potential around any closed loop must be zero.
14. The Wheatstone bridge is an arrangement of four resistances – \( R_1 \), \( R_2 \), \( R_3 \), \( R_4 \) as shown in the text. The null-point condition is given by \[ \frac{R_{1}}{R_{2}}=\frac{R_{3}}{R_{4}} \] using which the value of one resistance can be determined, knowing the other three resistances.
15. The potentiometer is a device to compare potential differences. Since the method involves a condition of no current flow, the device can be used to measure potential difference; internal resistance of a cell and compare emf’s of two sources.

POINTS TO PONDER

1. Current is a scalar although we represent current with an arrow. Currents do not obey the law of vector addition. That current is a scalar also follows from it’s definition. The current \( I \) through an area of cross-section is given by the scalar product of two vectors: \[ I=\mathbf{j} \cdot \Delta \mathbf{S} \] where \( \mathbf{j} \) and \( \Delta\mathbf{S} \) are vectors
2. Refer to \( V-I \) curves of a resistor and a diode as drawn in the text. A resistor obeys Ohm’s law while a diode does not. The assertion that \( V = IR \) is a statement of Ohm’s law is not true. This equation defines resistance and it may be applied to all conducting devices whether they obey Ohm’s law or not. The Ohm’s law asserts that the plot of \( I \) versus \( V \) is linear i.e., \( R \) is independent of \( V \).
   
   Equation \( \mathbf{E}=\rho \mathbf{j} \) leads to another statement of Ohm’s law, i.e., a conducting material obeys Ohm’s law when the resistivity of the material does not depend on the magnitude and direction of applied electric field.
3. Homogeneous conductors like silver or semiconductors like pure germanium or germanium containing impurities obey Ohm’s law within some range of electric field values. If the field becomes too strong, there are departures from Ohm’s law in all cases.
4. Motion of conduction electrons in electric field \( \mathbf{E} \) is the sum of (i) motion due to random collisions and (ii) that due to \( \mathbf{E} \). The motion due to random collisions averages to zero and does not contribute to \( v_d \) (Chapter 11, Textbook of Class XI). \( v_d \) , thus is only due to applied electric field on the electron.
5. The relation \( \mathbf{j}=\rho \ mathbf{v} \) should be applied to each type of charge carriers separately. In a conducting wire, the total current and charge density arises from both positive and negative charges: \[ \begin{array}{l} \mathbf{j}=\rho_{+} \mathbf{v}_{+}+\rho_{-} \mathbf{v}_{-} \\ \rho=\rho_{+}+\rho_{-} \end{array} \] Now in a neutral wire carrying electric current, \[ \rho_{+}=-\rho_{-} \] Further, \( v_+ \sim 0 \) which gives \[ \begin{array}{l} \rho=0 \\ \mathbf{j}=\rho_{-} \mathbf{v} \end{array} \] Thus, the relation \( \mathbf{j}=\rho \mathbf{v} \) does not apply to the total current charge density.
6. Kirchhoff’s junction rule is based on conservation of charge and the outgoing currents add up and are equal to incoming current at a junction. Bending or reorienting the wire does not change the validity of Kirchhoff’s junction rule.