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The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3) °C^(-1) and that of germanium is -5 × 10^(-2) °C^(-1). A tungsten wore of resistance 100 ohm is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is ?
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The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3)...
To determine the value of resistance \( R \) for the germanium wire such that the total resistance of the series combination remains constant with temperature, we can use the temperature coefficients of resistance.
Understanding Temperature Coefficients
- The temperature coefficient of resistance for tungsten ( \( \alpha_T \) ) is \( 4.5 \times 10^{-3} \, °C^{-1} \).
- The temperature coefficient of resistance for germanium ( \( \alpha_G \) ) is \( -5 \times 10^{-2} \, °C^{-1} \).
Resistance Change with Temperature
- The resistance of a conductor changes with temperature according to the formula:
\[ R(T) = R_0 (1 + \alpha (T - T_0)) \]
where \( R_0 \) is the initial resistance, \( \alpha \) is the temperature coefficient, and \( T \) is the temperature.
Finding the Condition for Constant Resistance
- For the tungsten wire, the change in resistance with temperature is:
\[ R_T(T) = 100 \, (1 + 4.5 \times 10^{-3} (T - T_0)) \]
- For the germanium wire, the change in resistance is:
\[ R_G(T) = R \, (1 - 5 \times 10^{-2} (T - T_0)) \]
- The total resistance \( R_{total} \) is:
\[ R_{total}(T) = R_T(T) + R_G(T) \]
- For the combination to remain constant, the change in resistance due to tungsten must equal the opposite change due to germanium:
\[ 100 \cdot 4.5 \times 10^{-3} = R \cdot 5 \times 10^{-2} \]
Calculating R
- Rearranging gives:
\[ R = \frac{100 \cdot 4.5 \times 10^{-3}}{5 \times 10^{-2}} \]
\[ R = \frac{0.45}{0.05} = 9 \, \Omega \]
Conclusion
- The value of resistance \( R \) for the germanium wire to ensure that the total resistance does not change with temperature is \( 9 \, \Omega \).
Understanding Temperature Coefficients
- The temperature coefficient of resistance for tungsten ( \( \alpha_T \) ) is \( 4.5 \times 10^{-3} \, °C^{-1} \).
- The temperature coefficient of resistance for germanium ( \( \alpha_G \) ) is \( -5 \times 10^{-2} \, °C^{-1} \).
Resistance Change with Temperature
- The resistance of a conductor changes with temperature according to the formula:
\[ R(T) = R_0 (1 + \alpha (T - T_0)) \]
where \( R_0 \) is the initial resistance, \( \alpha \) is the temperature coefficient, and \( T \) is the temperature.
Finding the Condition for Constant Resistance
- For the tungsten wire, the change in resistance with temperature is:
\[ R_T(T) = 100 \, (1 + 4.5 \times 10^{-3} (T - T_0)) \]
- For the germanium wire, the change in resistance is:
\[ R_G(T) = R \, (1 - 5 \times 10^{-2} (T - T_0)) \]
- The total resistance \( R_{total} \) is:
\[ R_{total}(T) = R_T(T) + R_G(T) \]
- For the combination to remain constant, the change in resistance due to tungsten must equal the opposite change due to germanium:
\[ 100 \cdot 4.5 \times 10^{-3} = R \cdot 5 \times 10^{-2} \]
Calculating R
- Rearranging gives:
\[ R = \frac{100 \cdot 4.5 \times 10^{-3}}{5 \times 10^{-2}} \]
\[ R = \frac{0.45}{0.05} = 9 \, \Omega \]
Conclusion
- The value of resistance \( R \) for the germanium wire to ensure that the total resistance does not change with temperature is \( 9 \, \Omega \).
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The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3) °C^(-1) and that of germanium is -5 × 10^(-2) °C^(-1). A tungsten wore of resistance 100 ohm is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is ?
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The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3) °C^(-1) and that of germanium is -5 × 10^(-2) °C^(-1). A tungsten wore of resistance 100 ohm is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is ? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3) °C^(-1) and that of germanium is -5 × 10^(-2) °C^(-1). A tungsten wore of resistance 100 ohm is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is ? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3) °C^(-1) and that of germanium is -5 × 10^(-2) °C^(-1). A tungsten wore of resistance 100 ohm is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is ?.
The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3) °C^(-1) and that of germanium is -5 × 10^(-2) °C^(-1). A tungsten wore of resistance 100 ohm is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is ? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3) °C^(-1) and that of germanium is -5 × 10^(-2) °C^(-1). A tungsten wore of resistance 100 ohm is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is ? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The temperature coefficient of resistance of tungsten is 4.5 × 10^(-3) °C^(-1) and that of germanium is -5 × 10^(-2) °C^(-1). A tungsten wore of resistance 100 ohm is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is ?.
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