The permeability of a metal is $0.1256\ \text{T}\cdot\text{m}\cdot\text{A}^{-1}$. Its relative permeability will be ($\frac{\mu_0}{4\pi} = 10^{-7}\ \text{SI unit}$, $\pi = 3.14$)
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Notice the mathematical relationship between the digits! The value $0.1256$ is exactly $10^5$ times larger than $1.256 \times 10^{-6}$ (which is approximately $4\pi \times 10^{-7}$). Recognizing the shifting decimal places of $12.56$ avoids expanding the full exponents manually.
Step 1: Understanding the Question: Identify the numerical relationship between a decimal value and standard physical constants through order-of-magnitude recognition. Step 2: Key Formula or Approach: Recognize that 1.256 × 10⁻⁶ ≈ 4π × 10⁻⁷ (the permeability of free space μ₀). The given value 0.1256 is 10⁵ times larger. Step 3: Detailed Explanation: Shifting the decimal point: 0.1256 = 1.256 × 10⁻¹, which is 10⁵ × (1.256 × 10⁻⁶). Spotting this factor of 100,000 avoids manually expanding the full power-of-ten chain. The value 12.56 is simply 1.256 × 10¹, a familiar multiple of 4π. Recognizing these decimal shifts as multiples of familiar constants dramatically speeds up numerical reasoning. Step 4: Final Answer: The value 0.1256 is 10⁵ times 1.256 × 10⁻⁶.