Understanding the Physical Quantity

The question asks to identify the physical quantity that is numerically equal to the resistance of a substance when it is in a specific shape and size: a rod 1 meter long with a cross-sectional area of 1 square meter. This specific definition points towards an intrinsic property of the material itself, rather than a property of a particular object made from the material.

Analyzing the Options

Let's look at the given options and understand what each one represents:

• Charge: Charge is a fundamental property of matter that experiences a force when placed in an electromagnetic field. It is measured in Coulombs (C). This is not related to the resistance of a material in the way described.
• Conductance: Conductance is the ease with which an electric current flows through a material. It is the reciprocal of resistance. While related to resistance, its definition is not tied to a standard 1m length and 1 sq m area.
• Resistivity: Resistivity is an intrinsic property of a material that quantifies how strongly it resists the flow of electric current. It is defined based on a standard volume of the material.
• Resistance: Resistance is the opposition to the flow of electric current in a specific object or circuit component. It depends not only on the material but also on its shape and size (length and cross-sectional area).

Relationship Between Resistance and Resistivity

The resistance (\(R\)) of a uniform conductor is directly proportional to its length (\(L\)) and inversely proportional to its cross-sectional area (\(A\)). The constant of proportionality is the resistivity (\(\rho\)) of the material. The formula is:

\(R = \rho \frac{L}{A}\)

From this formula, we can rearrange to find resistivity:

\(\rho = R \frac{A}{L}\)

Defining Resistivity Based on Standard Dimensions

The question describes a specific scenario where the rod made of the substance has:

• Length (\(L\)) = 1 m
• Cross-sectional Area (\(A\)) = 1 sq m (\(1 \, m^2\))

If we substitute these values into the formula for resistivity:

\(\rho = R \frac{1 \, m^2}{1 \, m}\)

\(\rho = R \times 1 \, m\)

However, the definition in the question states that the *physical quantity* is *numerically equal* to the resistance of a rod with these dimensions. Let's look at the formula for resistance instead:

\(R = \rho \frac{L}{A}\)

Substituting \(L = 1\) m and \(A = 1 \, m^2\):

\(R = \rho \frac{1 \, m}{1 \, m^2}\)

\(R = \rho \times \frac{1}{m}\)

This gives \(R = \frac{\rho}{m}\), which means Resistance is numerically equal to \( \rho / m \). This doesn't quite match the definition given in the question. There seems to be a slight unit mismatch if interpreting the formula literally with units. Let's re-examine the standard definition of resistivity.

Resistivity (\(\rho\)) is defined as the resistance (\(R\)) of a conductor of unit length (\(L=1\)) and unit cross-sectional area (\(A=1\)). Its units are Ohm-meter (\(\Omega \cdot m\)). If \(L=1\) m and \(A=1 \, m^2\), the formula \(R = \rho \frac{L}{A}\) becomes:

\(R = \rho \frac{1 \, m}{1 \, m^2} = \rho \times \frac{1}{m}\)

This formula suggests \(R\) has units of \(\Omega / m\), which is incorrect for resistance. The standard formula \(R = \rho \frac{L}{A}\) assumes \(L\) is in meters, \(A\) in square meters, \(R\) in Ohms, and \(\rho\) in Ohm-meters. Let's check the units in the formula \(R = \rho \frac{L}{A}\) when \(L=1\) m and \(A=1 \, m^2\):

Units of \(R\) = Units of \(\rho \times \frac{\text{Units of } L}{\text{Units of } A}\)

\(\Omega = (\Omega \cdot m) \times \frac{m}{m^2}\)

\(\Omega = (\Omega \cdot m) \times \frac{1}{m}\)

\(\Omega = \Omega\)

The units match. So, when \(L=1\) m and \(A=1 \, m^2\), the formula is \(R = \rho \frac{1}{1}\), which simplifies to \(R = \rho\). This means the numerical value of resistance (\(R\)) under these specific conditions (unit length and unit cross-sectional area) is equal to the numerical value of the resistivity (\(\rho\)) of the material.

Conclusion based on the Definition

The question's description:

"the physical quantity of the substance which is numerically equal to the resistance of a rod of that substance which is 1 m long and 1 sq m in cross-section"

matches the definition of resistivity. Resistivity is a material property, and its value is numerically equal to the resistance of a standard cube or rod of that material with dimensions 1m x 1m x 1m (or 1m length and 1 sq m area). Resistance depends on the specific dimensions of the object, while resistivity is independent of dimensions and characterizes the material itself.

Therefore, the physical quantity described is Resistivity.

Revision Table: Electrical Properties

Additional Information on Resistivity and Conductivity

Resistivity (\(\rho\)) is a fundamental property of a material that indicates how strongly it resists electrical current. Materials with high resistivity are poor conductors (insulators), while materials with low resistivity are good conductors.

Conductivity (\(\sigma\)) is the reciprocal of resistivity (\(\sigma = 1/\rho\)). It measures how well a material conducts electric current. Materials with high conductivity are good conductors.

The units of conductivity are Siemens per meter (S/m) or Ohm per meter inverse (\(\Omega^{-1} \cdot m^{-1}\)).

Resistivity and conductivity are intensive properties, meaning they do not depend on the amount or shape of the material, only on the material type and its temperature.

Examples of Resistivity (approximate values at 20°C):

• Copper: \(1.68 \times 10^{-8} \, \Omega \cdot m\) (Good conductor)
• Nichrome: \(1.1 \times 10^{-6} \, \Omega \cdot m\) (Resistor material)
• Glass: \(10^{10} \, \Omega \cdot m\) to \(10^{14} \, \Omega \cdot m\) (Insulator)