A Toaster's Element

solved TemperatureDependenceOfResistance

Q. 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 $T_{0} = 27.0$ °C) is found to be 75.3 Ω $R_{0} = 75.3$ Ω. When the toaster is connected to a 230 V $V = 230$ V supply, the current settles, after a few seconds, to a steady value of 2.68 A $I = 2.68$ A. The temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10⁻⁴ °C⁻¹ $α = 1.70 \times 10^{−4}$ °C⁻¹. What is the steady temperature of the nichrome element, in °C?

Inputs

room

T_{0}

27 °C

cold resistance

R_{0}

75.3 Ω

supply

V

230 V

current

I

2.68 A

coefficient

α

0.00017 °C⁻¹

Outputs

temperature

T

How it's solved

Solved with TemperatureDependenceOfResistance

Given

$T_{0} = 27.0$ °C
$R_{0} = 75.3$ Ω
$V = 230$ V
$I = 2.68$ A
$α = 1.70 \times 10^{−4}$ °C⁻¹

Find $T$ .

Working

$T_{0} = 27.0$ °C $= 300.15$ K
$α = 1.70 \times 10^{−4}$ °C⁻¹ $= 1.7 \times 10^{−4}$ K⁻¹
ohms law(element) $V = I \cdot R$
$230 = 2.68 \cdot R$
$R = 85.8208955$ Ω
resistance grows with temperature(element) $R = R_{0} \cdot (1 + α \cdot (T - T_{0}))$
$85.8208955 = 75.3 \cdot (1 + 1.7 \times 10^{−4} \cdot (T - 300.15))$
$T = 848.9$ °C
$T = 1122.03075$ K $= 848.880753$ °C
$T \approx 849$ °Cto 3 significant figures

Conclusion

Hot, the element passes less current than Ohm's law at room temperature would give; how much its resistance has grown tells how hot it is.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :a_toaster_s_element do4  kind :numerical5  describes "An electric toaster uses nichrome for its heating element. When a negligibly small current passes " \6            "through it, its resistance at room temperature (%{room}) is found to be %{cold_resistance}. When the " \7            "toaster is connected to a %{supply} supply, the current settles, after a few seconds, to a steady value " \8            "of %{current}. The temperature coefficient of resistance of nichrome averaged over the temperature " \9            "range involved is %{coefficient}. %{temperature}, in °C?"10  given room: "27.0 °C"11  given cold_resistance: "75.3 Ω"12  given supply: "230 V"13  given current: "2.68 A"14  given coefficient: "1.70e-4 °C-1"15  asks temperature: "What is the steady temperature of the nichrome element"16end

solution.rb

1require_relative "../questions/a_toaster_s_element"2require_relative "../science/current_electricity/temperature_dependence_of_resistance"34solve :a_toaster_s_element, using: TemperatureDependenceOfResistance do5  objects :element67  take room: T_08  take cold_resistance: element.reference_resistance9  take supply: element.voltage10  take current: element.current11  take coefficient: element.temperature_coefficient1213  temperature = element.temperature14  answer temperature:, in: "°C", figures: 31516  conclusion "Hot, the element passes less current than Ohm's law at room temperature would give; how much " \17             "its resistance has grown tells how hot it is."18end

playground.rb

1require_relative "../solutions/a_toaster_s_element"23play_with :a_toaster_s_element do4  title "A Toaster's Element"5end

The laws it drew on

quantity	written	units	within	from
voltage	$V$	V	—	OhmsLaw
current	$I$	A	—	OhmsLaw
resistance	$R$	Ω	0 … ∞	OhmsLaw
length of the conductor	$l$	m	—	OhmsLaw
cross section	$A$	m²	—	OhmsLaw
resistivity	$ρ$	Ω m	—	OhmsLaw
temperature coefficient	$α$	K⁻¹	—	TemperatureDependenceOfResistance
reference resistance	$R_{0}$	Ω	—	TemperatureDependenceOfResistance
reference temperature	$T_{0}$	K	—	TemperatureDependenceOfResistance
temperature	$T$	K	0 … ∞	TemperatureDependenceOfResistance
voltage of ground	$V_{g}$	V	—
current of ground	$I_{g}$	A	—
resistance of ground	$R_{g}$	Ω	0 … ∞
voltage of element	$V$	V	—
current of element	$I$	A	—
resistance of element	$R$	Ω	0 … ∞
resistivity of ground	$ρ_{g}$	Ω m	—
length of the conductor of ground	$l_{g}$	m	—
cross section of ground	$A_{g}$	m²	—
resistivity of element	$ρ$	Ω m	—
length of the conductor of element	$l$	m	—
cross section of element	$A$	m²	—
reference resistance of ground	$R_{0,g}$	Ω	—
temperature coefficient of ground	$α_{g}$	K⁻¹	—
temperature of ground	$T_{g}$	K	0 … ∞
reference resistance of element	$R_{0}$	Ω	—
temperature coefficient of element	$α$	K⁻¹	—
temperature of element	$T$	K	0 … ∞

V_{c} = I_{c} \cdot R_{c}

ohms law

R_{c} = \frac{ρ_{c} \cdot l_{c}}{A_{c}}

resistance from resistivity

R_{c} = R_{0,c} \cdot (1 + α_{c} \cdot (T_{c} - T_{0}))

resistance grows with temperature

V_{g} = I_{g} \cdot R_{g}

ohms law(ground)

V = I \cdot R

ohms law(element)

R_{g} = \frac{ρ_{g} \cdot l_{g}}{A_{g}}

resistance from resistivity(ground)

R = \frac{ρ \cdot l}{A}

resistance from resistivity(element)

R_{g} = R_{0,g} \cdot (1 + α_{g} \cdot (T_{g} - T_{0}))

resistance grows with temperature(ground)

R = R_{0} \cdot (1 + α \cdot (T - T_{0}))

resistance grows with temperature(element)