Resistivity of a material of wire is `3 * 10^(-6) Omega-m` and resistance of a particular thickness and length of wire is `2 Omega`. If the diameter of the wire gets doubled then the resistivity will be

Options

A. `1.1 * 10^(-2) Omega-m`
B. `1.5 * 10^(-3) Omega-m`
C. `2 * 10^(-6) Omega-m`
D. `3 * 10^(-6) Omega-m`

Answer

To solve the problem, we need to understand the concept of resistivity and how it relates to the dimensions of the wire. Here’s a step-by-step solution:

### Step 1: Understand the definition of resistivity
Resistivity (ρ) is a property of the material that quantifies how strongly it resists the flow of electric current. It is independent of the dimensions of the wire (length and cross-sectional area).

### Step 2: Identify given values
- The resistivity of the material of the wire is given as:
\[ \rho = 3 \times 10^{-6} \, \Omega \cdot m \]
- The resistance of the wire is given as:
\[ R = 2 \, \Omega \]
- The diameter of the wire is doubled.

### Step 3: Recall the relationship between resistance, resistivity, length, and area
The resistance (R) of a wire can be expressed using the formula:
\[ R = \frac{\rho L}{A} \]
where:
- \( R \) = Resistance
- \( \rho \) = Resistivity
- \( L \) = Length of the wire
- \( A \) = Cross-sectional area of the wire

### Step 4: Understand the effect of doubling the diameter
When the diameter of the wire is doubled, the cross-sectional area (A) changes. The area of a wire with diameter \( d \) is given by:
\[ A = \frac{\pi d^2}{4} \]
If the diameter is doubled (new diameter \( d' = 2d \)), the new area \( A' \) becomes:
\[ A' = \frac{\pi (2d)^2}{4} = \frac{\pi \cdot 4d^2}{4} = \pi d^2 = 4A \]
Thus, the area increases by a factor of 4.

### Step 5: Analyze the impact on resistivity
Resistivity (ρ) is a material property and does not change with the dimensions of the wire. Therefore, even if the diameter of the wire is doubled, the resistivity remains the same.

### Conclusion
The resistivity of the wire remains:
\[ \rho = 3 \times 10^{-6} \, \Omega \cdot m \]

### Final Answer
The resistivity when the diameter of the wire is doubled is:
\[ \boxed{3 \times 10^{-6} \, \Omega \cdot m} \]

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