A Silver Wire Warmed

solved TemperatureDependenceOfResistance

Q. A silver wire has a resistance of 2.1 Ω $R_{0} = 2.1$ Ω at 27.5 °C $T_{0} = 27.5$ °C, and a resistance of 2.7 Ω $R = 2.7$ Ω at 100 °C $T = 100$ °C. Determine the temperature coefficient of resistivity of silver, in °C⁻¹?

Inputs

cool

R_{0}

2.1 Ω

cool temperature

T_{0}

27.5 °C

warm

R

2.7 Ω

warm temperature

T

100 °C

Outputs

coefficient

α

How it's solved

Solved with TemperatureDependenceOfResistance

Given

$R_{0} = 2.1$ Ω
$T_{0} = 27.5$ °C
$R = 2.7$ Ω
$T = 100$ °C

Find $α$ .

Working

$T_{0} = 27.5$ °C $= 300.65$ K
$T = 100$ °C $= 373.15$ K
resistance grows with temperature(warm) $R = R_{0} \cdot (1 + α \cdot (T - T_{0}))$
$2.7 = 2.1 \cdot (1 + α \cdot (373.15 - 300.65))$
$α = 0.003941$ °C⁻¹
$α = 0.0039408867$ K⁻¹ $= 0.0039408867$ °C⁻¹
$α \approx 0.0039$ °C⁻¹to 2 significant figures

Conclusion

The resistance grows by the same fraction for every degree, so the fraction it grew over the rise in temperature is the coefficient.

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_silver_wire_warmed do4  kind :numerical5  describes "A silver wire has a resistance of %{cool} at %{cool_temperature}, and a resistance of %{warm} at " \6            "%{warm_temperature}. %{coefficient}, in °C⁻¹?"7  given cool: "2.1 Ω"8  given cool_temperature: "27.5 °C"9  given warm: "2.7 Ω"10  given warm_temperature: "100 °C"11  asks coefficient: "Determine the temperature coefficient of resistivity of silver"12end

solution.rb

1require_relative "../questions/a_silver_wire_warmed"2require_relative "../science/current_electricity/temperature_dependence_of_resistance"34solve :a_silver_wire_warmed, using: TemperatureDependenceOfResistance do5  objects :warm67  take cool: warm.reference_resistance8  take cool_temperature: T_09  take warm: warm.resistance10  take warm_temperature: warm.temperature1112  coefficient = warm.temperature_coefficient13  answer coefficient:, in: "°C-1", figures: 21415  conclusion "The resistance grows by the same fraction for every degree, so the fraction it grew over the rise " \16             "in temperature is the coefficient."17end

playground.rb

1require_relative "../solutions/a_silver_wire_warmed"23play_with :a_silver_wire_warmed do4  title "A Silver Wire Warmed"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 warm	$V$	V	—
current of warm	$I$	A	—
resistance of warm	$R$	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 warm	$ρ$	Ω m	—
length of the conductor of warm	$l$	m	—
cross section of warm	$A$	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 warm	$R_{0}$	Ω	—
temperature coefficient of warm	$α$	K⁻¹	—
temperature of warm	$T$	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(warm)

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

resistance from resistivity(ground)

R = \frac{ρ \cdot l}{A}

resistance from resistivity(warm)

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(warm)