Question

A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to a steady value of 2.8 A. What is the steady temperature of the heating element if the room temperature is 27.0 °C? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10–4 °C–1.

Answer 1

answer : 867°C

at the room temperature 27°C , the resistance of the heating element is $R_{27^{\circ}}= \frac{230}{3.2}=71.875\Omega$

[ according to Ohm's law, V = IR , so , R = V/I ]

at the steady temperature t°C , the resistance , $R_t=\frac{230}{2.8}= 82.143\Omega$

now, use the relation, $R=R_0[1+\alpha(T-T_0)]$

here, $R=R_t = 82.143\Omega$

$R_0=R_{27^{\circ}}=71.875\Omega$

$\alpha=1.7\times10^{-4}/^{\circ}C$

now, 82.143 = 71.875 [1 + 1.7 × 10^-4(t - 27)]

⇒82.143 - 71.875 = 71.875 × 1.7 × 10^-4(t - 27)

⇒ 0.084 × 10^4 = t - 27

⇒ 840 + 27 = t

⇒ t = 867°C

hence, temperature = 867°C

Answer 2

$\huge \underline \mathrm \purple{Question↣}$‎

‎

A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to a steady value of 2.8 A. What is the steady temperature of the heating element if the room temperature is 27.0 °C? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10–4 °C^(-1)

$\huge \underline \mathrm \green{Answer↣}$‎

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In the given problem,‎

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The supply voltage is $\bold{V = 230 V}$ ‎

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The initial current drawn is $\bold{I_{1} = 3.2 A}$‎

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Consider the initial resistance to be $\bold{ R_{1}}$ , which can be found by the following relation :‎

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⠀⠀⠀⠀⠀⠀⠀⠀$\Large \boxed{ \bold{ \:R _{ 1 } = \frac{ V }{ I }}}$‎

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Substituting values, we get‎ ;

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⠀⠀⠀⠀⠀⠀⠀⠀$\Large\sf{ : \implies \: R _{ 1 } = \Large{ \frac{ 230 }{ 3.2 }}}$ ‎

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\Large\sf\red{: \implies \: R _{ 1 } = 71.87 Ω}$ ‎

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Value of current at steady state , $\bold{ I_{2} = 2.8 A}$ ‎

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Value of resistance at steady state $\bold {= R_{2}}$ ‎

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$\bold{R_{2}}$ can be calculated by the following equation :‎

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⠀⠀⠀⠀⠀⠀⠀⠀$\Large\sf{ : \implies \: R _{ 2} = \Large{\frac{ 230 }{ 2.8 }}}$ ‎

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\Large\sf\red{: \implies \: R _{ 2 } = 81.14Ω}$

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The temperature coefficient of nichrome averaged over the temperature range involved is $\bold{ 1.70 x 10^{– 4} ° C^{ – 1}}$ ‎

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Value of initial temperature of nichrome , $\bold{T_{1} = 27.0 ° C}$ ‎

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Value of steady state temperature reached by nichrome $\bold { = T_{2}}$‎

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This temperature $\bold {T_{2}}$ can be obtained by the following formula :

⠀‎

‎⠀⠀⠀⠀$\boxed{\boxed{ \Large\sf{ \alpha = { \frac{ R _{ 2 } – R _{ 1 }}{ R _{ 1 } \left ( T _{ 2 } – T _{ 1 }\right )}}}}}$‎

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⠀⠀⠀⠀⠀⠀⠀⠀$\bold{ \to \: T_{2} - 27 = \Large{\frac{82.14 - 71.87}{71.87(1.7 \times 10^{ - 4})}} }$‎

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\bold{ \to \: T_{2} - 27 = 840.5}$‎

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\bold{ \to \: T_{2} = 840.5 + 27}$‎

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\large{\bold\red{ \to \: T_{2} = {867.5 ° C}}}$‎

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ʜᴇɴᴄᴇ, ᴛʜᴇ sᴛᴇᴀᴅʏ ᴛᴇᴍᴘᴇʀᴀᴛᴜʀᴇ ᴏғ ᴛʜᴇ ʜᴇᴀᴛɪɴɢ ᴇʟᴇᴍᴇɴᴛ ɪs 867.5 °C