Derive the expression for the current density of a conductor in terms of the conductivity and applied electric field. Explain, with reason how the mobility of electrons in a conductor changes when the potential difference applied is doubled, keeping the temperature of the conductor constant.

What is relaxation time ? Derive an expression for resistivity of a wire in terms of number density of free electrons and relaxation time.

Two cells of emfs E₁ & E₂ and internal resistances r₁ & r₂ respectively are connected in parallel. Obtain expressions for the equivalent. (i) resistance and (ii) emf of the combination

First a set of n equal resistors of R each is connected in series to a battery of emf E and internal resistance R. A current I is observed to flow. Then the n resistors are connected in parallel to the same battery. It is observed that the current becomes 10 times. What is n ?

The following table gives the length of three copper wires, their diameters, and the applied potential difference across their ends. Arrange the wires in increasing order according to the following : (i) The magnitude of the electric field within them, (ii) The drift speed of electrons through them, and (iii) The current density within them.

(i) The potential difference applied across a given resistor is altered so that the heat produced per second increases by a factor of 9. By what factor does the applied potential difference change ? (ii) In the figure shown, an ammeter A and a resistor of 4 W are connected to the terminals of the source. The emf of the source is 12 V having an internal resistance of 2 W. Calculate the voltmeter and ammeter readings.

Define relaxation time of the free electrons drifting in a conductor. How is it related to the drift velocity of free electrons ? Use this relation to deduce the expression for the electrical resistivity of the material.

Two material bars A and B of equal area of crosssection, are connected in series to a DC supply. A is made of usual resistance wire and B of an n-type semiconductor. (i) In which bar is drift speed of free electrons greater? (ii) If the same constant current continues to flow for a long time, how will the voltage drop across A and B be affected? Justify each error.

(a) Define the term ‘conductivity’ of a metallic wire. Write its SI unit. (b) Using the concept of free electrons in a conductor, derive the expression for the conductivity of a wire in terms of number density and relaxation time. Hence obtain the relation between current density and the applied electric field E.

Define the term current density of a metallic conductor. Deduce the relation connecting current density (J) and the conductivity σ of the conductor, when an electric field E, is applied to it.

(i) Derive an expression for drift velocity of free electrons. (ii) How does drift velocity of electrons in a metallic conductor vary with increase in temperature ? Explain.

(i) Deduce the relation between current I flowing through a conductor and drift velocity $v_{d}$ of the electrons. (ii) Figure shows a plot of current ‘I’ flowing through the cross-section of a wire versus the time ‘t’. Use the plot to find the charge flowing in 10 s through the wire.

The temperature coefficient of resistivity, for two materials A and B, are 0.0031 / °C and 0.0068 / °C respectively. Two resistors R1 and R2, made from materials A and B, respectively, have resistances of 200 Ω and 100 Ω at 0°C. Show on a diagram, the 'colour code', of a carbon resistor, that would have a resistance equal to the series combination of R1 and R2, at a temperature of 100°C. (Neglect the ring corresponding to the tolerance of the carbon resistor).

A student connects a cell, of emf E2 and internal resistance r2 with a cell of emf E1 and internal resistance r1, such that their combination has a net internal resistance less than r1. This combination is then connected across a resistance R. Draw a diagram of the 'set-up' and obtain an expression for the current flowing through the resistance.

A cell of emf ‘E’ and internal resistance ‘r’ is connected across a variable load resistor R. Draw the plots of the terminal voltage V versus (i) R and (ii) the current I. It is found that when R = 4 Ω, the current is 1 A when R is increased to 9 Ω, the current reduces to 0.5 A. Find the values of the emf E and internal resistance r.