Current Density in a Linear Conductor of Non-Uniform Cross-Section

The current density in a conductor is defined as the amount of current flowing per unit area of the cross-section. In other words, it gives us an idea of how concentrated the current is within the conductor.

When a steady current passes through a linear conductor with a non-uniform cross-section, the current density is inversely proportional to the area of the cross-section. This means that as the area of the cross-section decreases, the current density increases, and vice versa. The correct answer is option 'C': inversely proportional to the area of the cross-section.

Explanation:

1. Definition of Current Density:
Current density (J) is defined as the ratio of the magnitude of the current (I) passing through a conductor to the area (A) of its cross-section. Mathematically, it can be expressed as J = I/A.

2. Relationship with Area of Cross-Section:
In a linear conductor, the cross-sectional area is not constant but varies along its length. Let's consider a small section of the conductor with a smaller cross-sectional area (A1) and another section with a larger cross-sectional area (A2).

3. Conservation of Charge:
According to the principle of conservation of charge, the total amount of charge that enters a section of the conductor must be equal to the total amount of charge that leaves that section. In other words, the current flowing into a section must be equal to the current flowing out of it.

4. Variation of Current with Cross-Sectional Area:
Since the current is constant throughout the conductor, the product of current density and the area of the cross-section should remain constant along the length of the conductor. Mathematically, J1 * A1 = J2 * A2.

5. Inverse Proportionality:
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Conclusion:

In a linear conductor of non-uniform cross-section, the current density is inversely proportional to the area of the cross-section. This means that as the cross-sectional area decreases, the current density increases, and vice versa.