A metal wire has a resistance of 8 ohm at a temperature of 20 degree celcius . If the same wire has a resistance of 8.40 ohm at 70 degree celcius , what is the resistance of this wire in ohms when its temperature is -10 degree celcius ?
sorry I did it
To solve this problem, we can use the concept of temperature coefficient of resistance. The resistance of a wire changes with temperature due to the change in the wire's dimensions and resistance. The formula to calculate the resistance of a wire at a different temperature is:
R2 = R1 * (1 + α * (T2 - T1))
Where:
R1 = Resistance at temperature T1
R2 = Resistance at temperature T2
α = Temperature coefficient of resistance
T1 = Initial temperature
T2 = Final temperature
In this case, we need to find the resistance of the wire at -10 degrees Celsius.
Given:
R1 = 8 ohm (at 20 degrees Celsius)
T1 = 20 degrees Celsius
T2 = -10 degrees Celsius
Now, we need to calculate α, the temperature coefficient of resistance. The temperature coefficient of resistance is given by the formula:
α = (R2 - R1) / (R1 * (T2 - T1))
Substituting the given values:
α = (8.40 ohm - 8 ohm) / (8 ohm * (70 degrees Celsius - 20 degrees Celsius))
Simplifying the equation:
α = 0.40 ohm / (8 ohm * 50 degrees Celsius)
α = 0.40 ohm / 400 ohm.degrees Celsius
α = 0.001 ohm/degree Celsius
Now, substitute the values into the resistance formula:
R2 = 8 ohm * (1 + 0.001 ohm/degree Celsius * (-10 degrees Celsius - 20 degrees Celsius))
Simplifying the equation:
R2 = 8 ohm * (1 + 0.001 ohm/degree Celsius * (-30 degrees Celsius))
R2 = 8 ohm * (1 + 0.001 ohm/degree Celsius * -30 degrees Celsius)
R2 = 8 ohm * (1 + (-0.03 ohm))
R2 = 8 ohm * (0.97)
R2 = 7.76 ohm
Therefore, the resistance of the wire at a temperature of -10 degrees Celsius is approximately 7.76 ohms.