A Toaster's Element

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) is found to be 75.3 Ω. 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. The temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10⁻⁴ °C⁻¹. What is the steady temperature of the nichrome element, in °C? expects 849 °C

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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?

The playground

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

The solution

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 Ruby

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

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

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

The laws

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)