(1).The resistance of wire is giving by the relation Rt=Ro(1 alpha t Bt^2)
T is the temperature
alpha & Bt^2(beta) are constant
R=Resistance of the wire in ohms
Assuming the resistance of plantimum wire at 0°c,100°c,444.6°c and found to be 4.5,6.5 and 16.5 ohms respectively,Find the value of alpha & beta?
To find the values of alpha (α) and beta (β) in the given relation:
1. First, we need to plug in the resistance values (R) and the corresponding temperatures (T) into the equation Rt = Ro(1 + αT + βT^2), where Ro is the resistance at 0°C.
Given data:
- Resistance at 0°C (Ro) = 4.5 Ω
- Resistance at 100°C (R1) = 6.5 Ω
- Resistance at 444.6°C (R2) = 16.5 Ω
Plugging in these values into the equation, we get three equations:
Equation 1: R1 = Ro(1 + α * 100 + β * 100^2)
Equation 2: R2 = Ro(1 + α * 444.6 + β * 444.6^2)
2. Now, we can solve these equations simultaneously to find the values of α and β.
Subtracting Equation 1 from Equation 2:
R2 - R1 = Ro(α * (444.6 - 100) + β * (444.6^2 - 100^2))
16.5 - 6.5 = 4.5(α * 344.6 + β * 44054.84)
10 = 4.5α * 344.6 + 4.5β * 44054.84
Divide both sides by 4.5:
10/4.5 = α * 344.6 + β * 44054.84
2.222 = 344.6α + 976.43β .........(Equation 3)
3. Next, let's use Equation 1 to find α:
6.5 = Ro(1 + α * 100 + β * 100^2)
6.5 = 4.5(1 + α * 100 + β * 100^2)
Divide both sides by 4.5:
6.5/4.5 = 1 + α * 100 + β * 100^2
1.44 = 1 + α * 100 + β * 10000
Subtracting 1 from both sides:
0.44 = α * 100 + β * 10000
Let's call this equation (Equation 4).
4. Now, we have two equations with two variables (Equation 3 and Equation 4). Solve the system of equations to find the values of α and β.
Solving Equation 4 for α:
0.44 = α * 100 + β * 10000
α = (0.44 - β * 10000) / 100
Substituting this value of α in Equation 3:
2.222 = 344.6[(0.44 - β * 10000) / 100] + 976.43β
Simplify the equation:
2.222 = 3.446 - 34.46β + 976.43β
Combine like terms:
2.222 = 942.973β - 30.014β
Rearrange the equation:
932.959β = 2.222
Divide both sides by 932.959:
β ≈ 0.002378
5. Now, substitute the value of β back into Equation 4 to solve for α:
0.44 = α * 100 + 0.002378 * 10000
Simplify the equation:
0.44 = α * 100 + 23.78
Subtract 23.78 from both sides:
-23.34 = α * 100
Divide both sides by 100:
α ≈ -0.2334
Therefore, the approximate values of α and β are:
α ≈ -0.2334
β ≈ 0.002378