`'rho'` the electrical resistivity has dimensions

Options

A. `M^(1)L^(2)T^(-2)A^(-2)`
B. `M^(1)L^(3)T^(-1)A^(-2)`
C. `M^(1)L^(-2)T^(-3)A^(-2)`
D. `M^(1)L^(2)T^(-2)A^(-3)`

Answer

To find the dimensions of electrical resistivity (`ρ`), we can follow these steps:

### Step 1: Understand the relationship between resistance, resistivity, length, and area.
The formula for resistance (`R`) in terms of resistivity (`ρ`) is given by:
\[ R = \frac{\rho \cdot L}{A} \]
Where:
- `R` = Resistance
- `ρ` = Electrical resistivity
- `L` = Length of the conductor
- `A` = Cross-sectional area of the conductor

### Step 2: Rearrange the formula to express resistivity.
From the formula above, we can rearrange it to express resistivity:
\[ \rho = \frac{R \cdot A}{L} \]

### Step 3: Identify the dimensions of the known quantities.
We need to find the dimensions of each quantity in the equation:
- **Resistance (`R`)**: We can derive the dimensions of resistance using the formula for heat (energy) related to current:
\[ H = I^2 \cdot R \cdot T \]
Rearranging gives:
\[ R = \frac{H}{I^2 \cdot T} \]
The dimensions of heat (energy) are:
\[ [H] = [E] = [M L^2 T^{-2}] \]
The dimensions of current (`I`) are:
\[ [I] = [A] \]
The dimensions of time (`T`) are:
\[ [T] = [T] \]

Thus, substituting these into the resistance formula:
\[ [R] = \frac{[M L^2 T^{-2}]}{[A^2] \cdot [T]} = [M L^2 T^{-3} A^{-2}] \]

### Step 4: Substitute the dimensions into the resistivity formula.
Now we substitute the dimensions of `R`, `A`, and `L` into the rearranged formula for resistivity:
- The dimensions of area (`A`) are:
\[ [A] = [L^2] \]
- The dimensions of length (`L`) are:
\[ [L] = [L] \]

Substituting these into the resistivity formula:
\[ [ρ] = \frac{[M L^2 T^{-3} A^{-2}] \cdot [L^2]}{[L]} \]

### Step 5: Simplify the expression.
Now we simplify the expression:
\[ [ρ] = \frac{[M L^2 T^{-3} A^{-2}] \cdot [L^2]}{[L]} = [M L^3 T^{-3} A^{-2}] \]

### Conclusion: Final dimensions of electrical resistivity.
Thus, the dimensions of electrical resistivity (`ρ`) are:
\[ [ρ] = [M L^3 T^{-3} A^{-2}] \]