A resistor is to have a constant resistance of 30(ohm), independent of temperature. For this, an aluminum resistor with resistance R1 at 0(Celsius) is used in series with a carbon resistor with resistance R2 at 0(Celsius). Evaluate R1 and R2, given that a1=3.9 x 10^-3 degree Celsius^-1 for aluminum and a2=-0.5 x 10^-3degree Celsius^-1 for carbon.
i really don't know how to solve this, so pls. help me...Thanks in advance.
To solve this problem, we need to set up an equation that represents the condition for having a constant resistance of 30 Ohms, independent of temperature.
Let's assume R1 is the resistance of the aluminum resistor at 0 degrees Celsius, and R2 is the resistance of the carbon resistor at 0 degrees Celsius.
Since the resistance of a material changes with temperature, we need to take into account the temperature coefficient of resistance (denoted as α). The equation relating the change in resistance to change in temperature is:
ΔR = α * R * ΔT,
where ΔR is the change in resistance, α is the temperature coefficient of resistance, R is the initial resistance, and ΔT is the change in temperature.
For the aluminum resistor:
ΔR1 = α1 * R1 * ΔT,
For the carbon resistor:
ΔR2 = α2 * R2 * ΔT.
The total change in resistance is given as zero since we want to have a constant resistance independent of temperature. Therefore:
0 = ΔR1 + ΔR2 = α1 * R1 * ΔT + α2 * R2 * ΔT.
Since this equation must hold true for any temperature change, we can set ΔT to be any value. Let's choose ΔT = 1 degree Celsius for simplicity.
0 = α1 * R1 * 1 + α2 * R2 * 1,
0 = α1 * R1 + α2 * R2.
Now, we can substitute the values of α1 and α2 given in the problem:
0 = (3.9 * 10^-3) * R1 + (-0.5 * 10^-3) * R2.
Simplifying the equation, we can multiply both sides by 1000 to eliminate the scientific notation:
0 = 3.9 * R1 - 0.5 * R2.
Next, we need to deal with the fact that the total resistance when the two resistors are in series must be 30 Ohms:
R = R1 + R2 = 30 Ohms.
Now, we have two equations with two unknowns:
1) 0 = 3.9 * R1 - 0.5 * R2,
2) R = R1 + R2.
We can solve these equations simultaneously to find the values for R1 and R2.
From Equation 2, we can rewrite R2 in terms of R1:
R2 = R - R1.
Substituting this into Equation 1:
0 = 3.9 * R1 - 0.5 * (R - R1).
Simplifying:
0 = 3.9 * R1 - 0.5 * R + 0.5 * R1,
0 = 4.4 * R1 - 0.5 * R.
Now, we can solve for R1:
4.4 * R1 = 0.5 * R,
R1 = (0.5 * R) / 4.4.
Since R = R1 + R2, we can substitute R2 = R - R1:
R = R1 + (R - R1),
R = R.
This implies that R can be any value and, from Equation 2, that R2 = R - R1. Therefore, there are infinitely many possible solutions for R1 and R2 that will satisfy the condition.
To summarize, we found that there are no specific values for R1 and R2 that satisfy the condition of a constant resistance of 30 Ohms independent of temperature when using an aluminum resistor with α1 = 3.9 * 10^-3 degree Celsius^-1 and a carbon resistor with α2 = -0.5 * 10^-3 degree Celsius^-1.