Question

The permeability of a metal is $0.1256\ \text{T}\cdot\text{m}\cdot\text{A}^{-1}$. Its relative permeability will be ($\frac{\mu_0}{4\pi} = 10^{-7}\ \text{SI unit}$, $\pi = 3.14$)

• A. $10^5$
• B. $3 \times 10^5$
• C. $2 \times 10^6$
• D. $10^4$

Hint:

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.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the absolute magnetic permeability ($\mu$) of a specific metal. We need to compute its dimensionless relative permeability ($\mu_r$) using the constant value provided for the permeability of free space ($\mu_0$).

Step 2: Key Formula or Approach:
The relative permeability ($\mu_r$) of a material is defined as the ratio of its absolute magnetic permeability ($\mu$) to the permeability of free space ($\mu_0$): $$\mu_r = \frac{\mu}{\mu_0}$$ From the given constant $\frac{\mu_0}{4\pi} = 10^{-7}$, we can isolate $\mu_0$ as: $$\mu_0 = 4\pi \times 10^{-7}\ \text{T}\cdot\text{m}\cdot\text{A}^{-1}$$

Step 3: Detailed Explanation:
Let's first calculate the numerical value of $\mu_0$ using $\pi = 3.14$: $$\mu_0 = 4 \times 3.14 \times 10^{-7} = 12.56 \times 10^{-7}\ \text{T}\cdot\text{m}\cdot\text{A}^{-1}$$ The absolute permeability of the metal is given as $\mu = 0.1256\ \text{T}\cdot\text{m}\cdot\text{A}^{-1}$. Let's express this value in scientific notation to simplify the division: $$\mu = 0.1256 = 12.56 \times 10^{-2}\ \text{T}\cdot\text{m}\cdot\text{A}^{-1}$$ Now, substitute both values into the relative permeability ratio formula: $$\mu_r = \frac{12.56 \times 10^{-2}}{12.56 \times 10^{-7}}$$ The coefficient $12.56$ cancels out perfectly from the numerator and denominator: $$\mu_r = \frac{10^{-2}}{10^{-7}} = 10^{-2 - (-7)} = 10^{-2 + 7} = 10^5$$

Step 4: Final Answer:
The relative permeability of the metal is $10^5$, which corresponds to option (A).