3.17 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 is ∝ I, or V = R I, where R is called the resistance of the substance. The unit of resistance is ohm, represented by Ω: 1 Ω = 1 V/A.

4. The resistance R of a conductor depends on its length l and constant cross-sectional area A through the relation,


R = ρ l/A


where ρ, called resistivity is a property of the material and depends on temperature and pressure.

5. Electrical resistivity of different substances varies over a very wide range.


Metals have low resistivity, in the range of 10^−8 Ω m to 10^−6 Ω m.


Insulators like glass and rubber have 10^22 to 10^24 times greater resistivity.


Semiconductors like S i and G e 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 j vector gives the amount of charge flowing per second per unit area normal to the flow,j vector = n q vd vector


where n is the number density (number per unit volume) of charge carriers each of charge q, and vd vector is the drift velocity of the charge carriers. For electrons q = −e. If j vector is normal to a cross-sectional area A and is constant over the area, the magnitude of the current I through the area is


n e vd A.

8. Using electric field E = V/l, current I = n e vd A, and Ohm’s law, we obtain


eE/m = ρ (n e^2/m) vd

The proportionality between the force eE on the electrons in a metal due to the external field E and the drift velocity vd (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 τ,

vd = a τ = eEτ/m

where a is the acceleration of the electron. This gives

ρ = m/(ne^2τ)

9. In the temperature range in which resistivity increases linearly with temperature, the temperature coefficient of resistivity α 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


(a) V depends on I non-linearly.


(b) the relation between V and I depends on the sign of V for the same absolute value of V.


(c) The relation between V and I is non-unique.

An example of


(a) is when ρ increases with I (even if temperature is kept fixed).


A rectifier combines features (a) and (b).


GaAs (gallium arsenide) shows the feature (c).

11. When a source of EMF ε is connected to an external resistance R, the voltage V ext across R is given by


V ext = IR = {ε/(R + r)} × R


where r is the internal resistance of the source.

12.


(a) Total resistance R of n resistors connected in series is given by


R = R1 + R2 +..... + Rn


(b) Total resistance R of n resistors connected in parallel is given by


1/R = 1/R1 + 1/R2 +..... + 1/Rn

13. Kirchhoffs Rules -

(a) Junction Rule: At any junction of circuit elements, the sum of currents entering the junction must be equal to the sum of currents leaving it.


(b) 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 - R1, R2 , R3, R4 as shown in the text. The null-point condition is given by


R1/R2 = R3/R4


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 EMFs of two sources.