Question

Example 3.3 An electric toaster uses nichrome for its heating
element. When a negligibly small current passes through it, its
resistance at room temperature (27.0 °C) is found to be 75.3 22. 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 x 10-4 °C-1.​

Answer 1

Answer:

Steady temperature = 848.7647 ° C

Explanation:

Given:

Temperature (T₀) = 27° C

Resistance (R₀) = 75.3 Ω

Voltage = 230 V

Current = 2.68 A

Temperature coefficient of resistance of nichrome wire = 1.70 × 10⁻⁴ C⁻¹

To Find:

Steady temperature of nichrome element (T)

Solution:

First let us find the resistance R at temperature T

By Ohm's law we know that

R = V/I

where R is the resistance, V is the voltage and I is the current

Substitute the data,

R = 230/2.68

R = 85.82 Ω

Hence the resistance of the wire at temperature T is 85.82 Ω.

Also resistance of a conductor at t° C is given by,

$\sf R=R_0(1+ \alpha (T-T_0))$

where R is the resistance at temperature T

R₀ is the resistance at temperature T₀

α is the coefficient of resistance

Substitute the data,

$\sf 85.82=75.3(1+ 1.70\times 10^{-4} (T-27))$

$\sf (1 + 1.7\times 10^{-4} T-45.9\times 10^{-4} )=1.1397$

$\sf 0.99541+1.7\times 10^{-4} T=1.1397$

$\sf 1.7\times 10^{-4} T=0.14429$

$\sf T =\dfrac{0.14429\times 10^4}{1.7}$

$\sf T = 848.7647 \:^{o} C$

Hence the steady temperature of nichrome element is 848.7647° C.

Answer 2

$\Large\bf{\color{indigo}GiVeN,}$

• When a negligible small current passes through an electric toaster, it's resistance at room temp. (27.0 °C) is $\bf{75.3\:\Omega}$.

• 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.

☆ According to Ohm's law,

$\orange\bigstar\:\:\bf\green{Resistance\:=\:\dfrac{Voltage}{Current}\:}$

$\longmapsto\:\:\bf{Resistance\:=\:\dfrac{230}{2.68}\:}$

$\longmapsto\:\:\bf{Resistance\:=\:85.82\:\Omega\:}$

$\bf\red{We\:know\:that,}$

☆ At particular temp., the resistance variation is given as,

$\green\bigstar\:\:\bf\blue{R\:=\:R_o\:\Big[1\:+\:\alpha\:{(T\:-\:T_o)}\Big]\:}$

$\bf\pink{Where,}$

• R is the resistance at a given temperature, i.e. $\bf{85.82\:\Omega}$.

• $\bf{R_o}$ is the resistance at room temperature, i.e. $\bf{75.3\:\Omega}$.

• $\bf{\alpha}$ is the temp. coefficient of resistance, i.e. $\bf{1.7\times{10^{-4}}\:/°C}$.

• T is the given temperature.

• $\bf{T_o}$ is the room temperature, i.e. 27°C.

$:\implies\:\:\bf{85.82\:=\:75.3\:\Big[1\:+\:1.7\times{10^{-4}}\:{(T\:-\:27)}\Big]\:}$

$:\implies\:\:\bf{\dfrac{85.82}{75.3}\:=\:1\:+\:1.7\times{10^{-4}}\:{(T\:-\:27)}\:}$

$:\implies\:\:\bf{1.13\:-\:1\:=\:1.7\times{10^{-4}}\:{(T\:-\:27)}\:}$

$:\implies\:\:\bf{(T\:-\:27)\:=\:\dfrac{0.13}{1.7\times{10^{-4}}}}$

$:\implies\:\:\bf\green{T\:=\:848.81°C}$