Power in AC Circuits: Practice Problem & Solution
Two cables of copper are of equal lengths. One of them has a single wire of area of cross-section A, while other has 10 wires of cross-sectional area A/10 each. Give their suitability for transporting A.C. and D.C (1994)
Short Solution (Best for NEET):
To solve this problem, we apply the core principles of Power in AC Circuits. Understanding the underlying formula is key to arriving at the correct answer below:
Alternating current (A.C.) exhibits the skin effect, concentrating on the conductor's surface.
Multiple stranded wires have a larger effective surface area, reducing resistance for A.C.
Direct current (D.C.) flows uniformly across the entire cross-section.
Thus, multiple strands are suitable for A.C., and either can be used for D.C.
Detailed Solution (Best of Understanding Complex Concepts)
Core Concepts Involved
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Skin Effect (for A.C.): When an Alternating Current (A.C.) flows through a conductor, it does not distribute uniformly across the cross-section. The changing magnetic field causes the current density to be highest at the surface (or "skin") of the conductor and to decrease deeper inside. Because the current is restricted to a smaller effective area, the A.C. resistance of the wire is higher than its D.C. resistance.
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Uniform Current Distribution (for D.C.): Direct Current (D.C.) flows steadily in one direction and creates no changing magnetic fields. Therefore, D.C. distributes uniformly across the entire cross-sectional area of the conductor.
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Surface Area to Volume Ratio: A bundle of thinner wires has a significantly larger total surface area (perimeter) compared to a single thick solid wire of the same total cross-sectional area.
Step-by-Step Solution
Step 1: Analyze the Geometry of the Two Cables
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Cable 1 (Solid): A single wire with a cross-sectional area of $A$.
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Cable 2 (Stranded): 10 thinner wires, each with an area of $A/10$. The total cross-sectional area is $10 \times \frac{A}{10} = A$.Both cables contain the exact same amount of copper per unit length.
Step 2: Evaluate Suitability for Direct Current (D.C.)
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D.C. flows uniformly through the entire available cross-section of the metal.
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Since both cables have the exact same total cross-sectional area ($A$) and are made of the same material (copper), they will offer the exact same electrical resistance to D.C.
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Result for D.C.: Both cables are equally suitable. You can use either.
Step 3: Evaluate Suitability for Alternating Current (A.C.)
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Due to the skin effect, A.C. prefers to travel along the outer surface of the conductor.
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In Cable 1 (single thick wire), the inner portion of the thick copper wire goes largely unused, making its effective resistance to A.C. unnecessarily high.
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In Cable 2 (multiple thin wires), the current can flow along the outer surface of each individual thin wire. The combined total surface area of the 10 smaller strands is significantly larger than the surface area of the single thick wire.
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This increased surface area gives the A.C. much more room to flow, drastically lowering the effective A.C. resistance and reducing energy loss (heat).
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Result for A.C.: The multiple-strand cable is much more efficient and suitable.
Final Conclusion:
The multiple-strand cable must be used for transporting A.C., whereas either cable can be used for transporting D.C. (matching Option 1).
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