A toroid has `2000` turns. The inner & outer radii of its core are `11cm` and `12cm` respectively. The magnetic field in the core for a current of `0*7A` is `2*5T`. What is relative permeability of core?

Answer

To find the relative permeability of the core of the toroid, we can follow these steps:

### Step 1: Calculate the Mean Radius
The mean radius \( r \) of the toroid can be calculated using the inner radius \( r_1 \) and the outer radius \( r_2 \):
\[ r = \frac{r_1 + r_2}{2} \]
Given:
- Inner radius \( r_1 = 11 \, \text{cm} = 0.11 \, \text{m} \)
- Outer radius \( r_2 = 12 \, \text{cm} = 0.12 \, \text{m} \)

Calculating:
\[ r = \frac{0.11 + 0.12}{2} = \frac{0.23}{2} = 0.115 \, \text{m} \]

### Step 2: Use the Formula for Magnetic Field in a Toroid
The magnetic field \( B \) in a toroid is given by the formula:
\[ B = \mu_0 \cdot \mu_r \cdot \frac{N \cdot I}{2 \pi r} \]
Where:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) (permeability of free space)
- \( \mu_r \) is the relative permeability (what we need to find)
- \( N \) is the number of turns (2000)
- \( I \) is the current (0.7 A)
- \( r \) is the mean radius (0.115 m)

### Step 3: Rearranging the Formula
Rearranging the formula to solve for \( \mu_r \):
\[ \mu_r = \frac{B \cdot 2 \pi r}{\mu_0 \cdot N \cdot I} \]

### Step 4: Substitute the Values
Substituting the known values into the rearranged formula:
\[ B = 2.5 \, \text{T}, \quad r = 0.115 \, \text{m}, \quad N = 2000, \quad I = 0.7 \, \text{A} \]
\[ \mu_r = \frac{2.5 \cdot 2 \pi \cdot 0.115}{(4\pi \times 10^{-7}) \cdot 2000 \cdot 0.7} \]

### Step 5: Simplifying the Expression
Calculating the numerator:
\[ \text{Numerator} = 2.5 \cdot 2 \cdot \pi \cdot 0.115 = 0.5775 \pi \]
Calculating the denominator:
\[ \text{Denominator} = (4\pi \times 10^{-7}) \cdot 2000 \cdot 0.7 = 5.6 \times 10^{-4} \pi \]

### Step 6: Final Calculation
Now substituting back:
\[ \mu_r = \frac{0.5775 \pi}{5.6 \times 10^{-4} \pi} = \frac{0.5775}{5.6 \times 10^{-4}} \approx 1031.25 \]

Thus, the relative permeability \( \mu_r \) is approximately \( 1031.25 \).

### Final Answer
\[ \mu_r \approx 1031.25 \]

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