3 Marks Question

Question 1013 Marks

An electric toaster uses nichrome for its heating element. When a negligibly small current passes through it, its resistance at room temperature $\left(27.0^{\circ} C \right)$ is found to be $75.3 \Omega$. When the toaster is connected to a $230 V$ supply, the current settles, after a few seconds, to a steady value of $2.68 A$. What is the steady temperature of the nichrome element? The temperature coefficient of resistance of nichrome averaged over the temperature range involved, is $1.70 \times 10^{-4}{ }^{\circ} C ^{-1}$.

Answer

When the current through the element is very small, heating effects can be ignored and the temperature $T_1$ of the element is the same as room temperature. When the toaster is connected to the supply, its initial current will be slightly higher than its steady value of $2.68 A$. But due to heating effect of the current, the temperature will rise. This will cause an increase in resistance and a slight decrease in current. In a few seconds, a steady state will be reached when temperature will rise no further, and both the resistance of the element and the current drawn will achieve steady values. The resistance $R_2$ at the steady temperature $T_2$ is
$
R_2=\frac{230 V }{2.68 A }=85.8 \Omega
$
Using the relation
$
R_2=R_1\left[1+\alpha\left(T_2-T_1\right)\right]
$
with $\alpha=1.70 \times 10^{-4}{ }^{\circ} C ^{-1}$, we get
$
T_2-T_1=\frac{(85.8-75.3)}{(75.3) \times 1.70 \times 10^{-4}}=820^{\circ} C
$
that is, $T_2=(820+27.0){ }^{\circ} C =847{ }^{\circ} C$
Thus, the steady temperature of the heating element (when heating effect due to the current equals heat loss to the surroundings) is $847^{\circ} C$.

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Question 1023 Marks

(a) In Example 3.1, the electron drift speed is estimated to be only a few $mm s ^{-1}$ for currents in the range of a few amperes? How then is current established almost the instant a circuit is closed?
(b) The electron drift arises due to the force experienced by electrons in the electric field inside the conductor. But force should cause acceleration. Why then do the electrons acquire a steady average drift speed?
(c) If the electron drift speed is so small, and the electron's charge is small, how can we still obtain large amounts of current in a conductor?
(d) When electrons drift in a metal from lower to higher potential, does it mean that all the 'free' electrons of the metal are moving in the same direction?
(e) Are the paths of electrons straight lines between successive collisions (with the positive ions of the metal) in the (i) absence of electric field, (ii) presence of electric field?

Answer

(a) Electric field is established throughout the circuit, almost instantly (with the speed of light) causing at every point a local electron drift. Establishment of a current does not have to wait for electrons from one end of the conductor travelling to the other end. However, it does take a little while for the current to reach its steady value.
(b) Each 'free' electron does accelerate, increasing its drift speed until it collides with a positive ion of the metal. It loses its drift speed after collision but starts to accelerate and increases its drift speed again only to suffer a collision again and so on. On the average, therefore, electrons acquire only a drift speed.
(c) Simple, because the electron number density is enormous, $\sim 10^{29} m ^{-3}$.
(d) By no means. The drift velocity is superposed over the large random velocities of electrons.
(e) In the absence of electric field, the paths are straight lines; in the presence of electric field, the paths are, in general, curved.

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Physics 3 Marks Question