A resistance thermometer has a resistance of 4.0\,\Omega at 0\,^\circ\text{C} and 6.0\,\Om… — JAMB UTME

A resistance thermometer has a resistance of $4.0\,\Omega$ at $0\,^\circ\text{C}$ and $6.0\,\Omega$ at $100\,^\circ\text{C}$ . The thermometer is connected in series with a fixed $10.0\,\Omega$ resistor across a $24.0\,\text{V}$ DC power source having an internal resistance of $1.0\,\Omega$ . If a steady current of $1.2\,\text{A}$ flows through the circuit, what is the temperature of the thermometer's environment?

$250\,^\circ\text{C}$ Answer
B
$300\,^\circ\text{C}$
C
$450\,^\circ\text{C}$
D
$350\,^\circ\text{C}$

Answer

The temperature of the thermometer environment is

250\,^\circ\text{C}

By applying the complete circuit equation

E = I(R_\theta + R_{\text{fixed}} + r)

, the total circuit resistance is found to be

20.0\,\Omega

. Subtracting the fixed resistance of

10.0\,\Omega

and the cell internal resistance of

1.0\,\Omega

yields the thermometer resistance

R_\theta = 9.0\,\Omega

. Substituting this into the thermometric relation

\theta = \frac{R_\theta - R_0}{R_{100} - R_0} \times 100\,^\circ\text{C}

gives

\frac{9.0 - 4.0}{6.0 - 4.0} \times 100 = 250\,^\circ\text{C}

Step-by-Step Solution

Calculate total circuit resistance using Ohm's Law and internal resistance equation

R_{\text{total}} = \frac{E}{I} = \frac{24.0\,\text{V}}{1.2\,\text{A}} = 20.0\,\Omega

The electromotive force (e.m.f) of the source equals total current multiplied by total resistance including internal resistance.

Determine the resistance of the thermometer at the unknown temperature (

R_\theta

)

R_\theta = R_{\text{total}} - R_{\text{fixed}} - r = 20.0\,\Omega - 10.0\,\Omega - 1.0\,\Omega = 9.0\,\Omega

The circuit components are in series, so total resistance is the sum of external resistances and internal resistance.

Apply the linear resistance thermometer temperature scale formula

$\theta = \frac{R_\theta - R_0}{R_{100} - R_0} \times 100\,^\circ\text{C} = \frac{9.0 - 4.0}{6.0 - 4.0} \times 100 = \frac{5.0}{2.0} \times 100 = 250\,^\circ\text{C}$

Resistance varies linearly with temperature between the ice point (

0\,^\circ\text{C}

) and steam point (

100\,^\circ\text{C}

Key Concept

Integration of Ohm's Law, internal resistance of a cell, and resistance thermometry

Estimated Time:2m 0s

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