Resistance of tungsten wire at `150^(@)C` is `133 Omega`. Its resistance temperature coefficient is `0.0045//^(@)C`. The resistance of this wire at `500^(@)C` will be

Options

A. `180Omega`
B. `225Omega`
C. `258Omega`
D. `317Omega`

Answer

To find the resistance of a tungsten wire at \(500^\circ C\) given its resistance at \(150^\circ C\) and the temperature coefficient of resistance, we can use the formula for resistance as a function of temperature:

\[ R_T = R_0 \left(1 + \alpha (T - 0)\right) \]

Where:
- \(R_T\) is the resistance at temperature \(T\),
- \(R_0\) is the resistance at \(0^\circ C\),
- \(\alpha\) is the temperature coefficient of resistance,
- \(T\) is the temperature in degrees Celsius.

### Step 1: Write the equation for resistance at \(150^\circ C\)

Given:
- \(R_{150} = 133 \, \Omega\)
- \(\alpha = 0.0045 \, \text{per} \, ^\circ C\)

Using the formula:

\[ R_{150} = R_0 \left(1 + \alpha \cdot 150\right) \]

Substituting the known values:

\[ 133 = R_0 \left(1 + 0.0045 \cdot 150\right) \]

### Step 2: Calculate \(1 + \alpha \cdot 150\)

Calculating \(0.0045 \cdot 150\):

\[ 0.0045 \cdot 150 = 0.675 \]

Thus,

\[ 1 + 0.675 = 1.675 \]

### Step 3: Substitute back to find \(R_0\)

Now we can rewrite the equation:

\[ 133 = R_0 \cdot 1.675 \]

To find \(R_0\):

\[ R_0 = \frac{133}{1.675} \approx 79.6 \, \Omega \]

### Step 4: Write the equation for resistance at \(500^\circ C\)

Now we need to find the resistance at \(500^\circ C\):

\[ R_{500} = R_0 \left(1 + \alpha \cdot 500\right) \]

Substituting \(R_0\) and \(\alpha\):

\[ R_{500} = 79.6 \left(1 + 0.0045 \cdot 500\right) \]

### Step 5: Calculate \(1 + \alpha \cdot 500\)

Calculating \(0.0045 \cdot 500\):

\[ 0.0045 \cdot 500 = 2.25 \]

Thus,

\[ 1 + 2.25 = 3.25 \]

### Step 6: Substitute back to find \(R_{500}\)

Now we can calculate \(R_{500}\):

\[ R_{500} = 79.6 \cdot 3.25 \]

Calculating this gives:

\[ R_{500} \approx 258.2 \, \Omega \]

### Final Answer

The resistance of the tungsten wire at \(500^\circ C\) is approximately \(258 \, \Omega\).