A Silver Wire Warmed

Q. A silver wire has a resistance of 2.1 Ω at 27.5 °C, and a resistance of 2.7 Ω at 100 °C. Determine the temperature coefficient of resistivity of silver, in °C⁻¹? expects 0.0039 °C⁻¹

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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?

The playground

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

The solution

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 Ruby

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

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

1require_relative "../solutions/a_silver_wire_warmed"23play_with :a_silver_wire_warmed do4  title "A Silver Wire Warmed"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 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)