Question

A wire of resistance 4 \(\Omega\) is used to make a coil of radius 7 cm. The wire has a diameter of 1.4 mm and the resistivity of its material is 2 x 10\(^{-7}\) \(\Omega\) m. The number of turns in the coil will be

• A. 70
• B. 40
• C. 140
• D. 20

Hint:

In such problems, break down the question into smaller parts. First, deal with the wire's intrinsic properties (resistance, resistivity, dimensions) to find the total length. Then, use that length in the context of the coil's geometry to find the required quantity (number of turns).

Answer 1

The Correct Option is A

Step 1: Conceptual Overview:
This problem requires integrating electrical and geometrical properties of a wire. We must first determine the wire's total length using its resistance and then use this length to calculate the number of turns in a coil.

Step 2: Governing Equations:
1. Wire Resistance: \(R = \rho \frac{L}{A}\) (where \(R\) = resistance, \(\rho\) = resistivity, \(L\) = length, \(A\) = cross-sectional area).2. Wire Cross-sectional Area: \(A = \pi r_{wire}^2\) (where \(r_{wire}\) = wire radius).3. Coil Wire Length: \(L = n \times (2\pi r_{coil})\) (where \(n\) = number of turns, \(r_{coil}\) = coil radius).

Step 3: Detailed Procedure:
Provided Data:
Total resistance, \(R = 4 \, \Omega\).
Coil radius, \(r_{coil} = 7 \, \text{cm} = 0.07 \, \text{m}\).
Wire diameter, \(d_{wire} = 1.4 \, \text{mm}\), thus wire radius, \(r_{wire} = 0.7 \, \text{mm} = 0.7 \times 10^{-3} \, \text{m}\).
Resistivity, \(\rho = 2 \times 10^{-7} \, \Omega \cdot \text{m}\).
Calculations:
Part 1: Determine Wire Length (L).
Calculate the wire's cross-sectional area (\(A\)):\[ A = \pi r_{wire}^2 = \pi (0.7 \times 10^{-3} \, \text{m})^2 = \pi (0.49 \times 10^{-6} \, \text{m}^2) \]Rearrange the resistance formula to solve for length \(L\):\[ L = \frac{R A}{\rho} \]Substitute values:\[ L = \frac{(4 \, \Omega) \times (\pi \times 0.49 \times 10^{-6} \, \text{m}^2)}{2 \times 10^{-7} \, \Omega \cdot \text{m}} \]\[ L = 9.8\pi \, \text{m} \]Part 2: Determine Number of Turns (n).
Calculate the circumference of one coil turn:\[ \text{Circumference} = 2 \pi r_{coil} = 2 \pi (0.07 \, \text{m}) = 0.14\pi \, \text{m} \]Calculate the number of turns by dividing total wire length by the circumference of one turn:\[ n = \frac{L}{2 \pi r_{coil}} \]\[ n = \frac{9.8\pi \, \text{m}}{0.14\pi \, \text{m}} = 70 \]

Step 4: Conclusion:
The coil contains 70 turns.