Three students Shweta, Ayesha and Samridhi were performing an experiment to understand the factors on which the resistance of a conductor depends. Each one of them completed an electric circuit with the help of a cell, an ammeter, a plug key and wire.

Shweta put a nichrome wire of length 'l' in the circuit and after plugging the key, noted the current in the ammeter.

Ayesha put a nichrome wire of the same thickness but twice the length, i.e. '2l', in the circuit and after plugging the key, noted the current in the ammeter.

Samridhi took a copper wire of length 'l' and the same thickness in the circuit and after plugging the key, noted the current in the ammeter.

(a) If the ammeter reading is X ampere with nichrome wire of length 'l', then what will be the ammeter reading if the length of the nichrome wire is doubled with the same area of cross-section?

(b) What happens to the ammeter reading if the area of cross-section of the nichrome wire is doubled, keeping the length of the wire 'l' the same?

(c) Define 'resistivity'. Write its SI unit. Compare the resistivity of an alloy with its constituent metals.

OR

(c) Give reason:

(i) Tungsten is used almost exclusively for making the filament of electric lamps.

(ii) Conductors of bread-toasters are made of an alloy rather than a pure metal.

(a) Length doubled ($l \to 2l$), same area, same nichrome material, same cell (emf unchanged):

From Eq. (11.10), $R=\rho\dfrac{l}{A}$, resistance is directly proportional to length, so doubling the length doubles the resistance: $R' = 2R$.

Since the same number of cells is used, the potential difference V applied is the same. By Ohm's law $I=V/R$ (Eq. 11.7), current is inversely proportional to resistance. Doubling R halves the current:

$$I' = \frac{V}{2R} = \frac{X}{2}\ \text{A}$$

So Ayesha's ammeter reads $X/2$ ampere — this matches the NCERT Activity 11.3 observation that "the ammeter reading decreases to one-half when the length of the wire is doubled."

(b) Area of cross-section doubled ($A \to 2A$), same length $l$, same nichrome wire:

From $R=\rho\dfrac{l}{A}$, resistance is inversely proportional to area, so doubling A halves the resistance: $R'' = R/2$.

By Ohm's law, with V unchanged, current doubles:

$$I'' = \frac{V}{R/2} = 2\times\frac{V}{R} = 2X\ \text{A}$$

This matches the text: "The ammeter reading is increased when a thicker wire … is used."

(c) Resistivity: As defined in Section 11.5, resistivity ($\rho$) is the constant of proportionality in $R=\rho\dfrac{l}{A}$; it is numerically the resistance of a conductor of unit length and unit cross-sectional area, and is a characteristic property of the material (independent of its dimensions). Its SI unit is $\Omega\,\text{m}$ (ohm-metre). From Table 11.2, an alloy's resistivity is higher than that of its constituent pure metals — e.g. nichrome (an alloy of Ni, Cr, Mn, Fe) has $\rho = 100\times10^{-6}\,\Omega\,\text{m}$, far higher than nickel ($6.84\times10^{-8}\,\Omega\,\text{m}$), chromium ($12.9\times10^{-8}\,\Omega\,\text{m}$) or iron ($10.0\times10^{-8}\,\Omega\,\text{m}$) individually. This higher resistivity is why alloys are preferred in heating elements — they also resist oxidation (burning) at high temperature.

OR (c) Reasons:

(i) Tungsten for filaments: The bulb filament must be heated to very high temperature so that it glows and emits light, without melting. Tungsten has an extremely high melting point ($3380^{\circ}\text{C}$), so it can withstand the intense heating produced by Joule heating ($H=I^2Rt$) without melting, unlike most other metals.

(ii) Alloy conductors in toasters: As given in Section 11.5 of the chapter, alloys such as nichrome have (a) higher resistivity than pure constituent metals, giving greater and more controllable heating for a given size of wire, and (b) they do not oxidise (burn) readily even at the high operating temperatures reached in a bread-toaster or electric iron, unlike pure metals which would corrode/burn out quickly.