Calculate the magnetic field intensity at a distance of `20cm.` from a pole of strength `40Am` in air. Find the magnetic induction at the same point.

Explanation

To solve the problem, we need to calculate the magnetic field intensity (H) and the magnetic induction (B) at a distance of 20 cm from a magnetic pole of strength 40 Am.

### Step-by-Step Solution:

**Step 1: Convert the distance from cm to meters.**
- Given distance = 20 cm = 0.20 m

**Step 2: Use the formula for magnetic field intensity (H) due to a magnetic pole.**
- The formula for magnetic field intensity (H) at a distance (r) from a magnetic pole of strength (m) is given by:
\[ H = \frac{m}{4\pi r^2} \]
- Here, \( m = 40 \, \text{Am} \) and \( r = 0.20 \, \text{m} \).

**Step 3: Substitute the values into the formula.**
\[ H = \frac{40}{4\pi (0.20)^2} \]

**Step 4: Calculate \( (0.20)^2 \).**
\[ (0.20)^2 = 0.04 \]

**Step 5: Substitute \( (0.20)^2 \) back into the equation for H.**
\[ H = \frac{40}{4\pi (0.04)} \]

**Step 6: Calculate \( 4\pi (0.04) \).**
\[ 4\pi (0.04) = 0.50265 \, \text{(approximately)} \]

**Step 7: Calculate H.**
\[ H = \frac{40}{0.50265} \approx 79.6 \, \text{A/m} \]

**Step 8: Calculate the magnetic induction (B).**
- The relationship between magnetic induction (B) and magnetic field intensity (H) is given by:
\[ B = \mu H \]
- In air, the permeability (\( \mu \)) is approximately equal to the permeability of free space (\( \mu_0 \)), which is \( 4\pi \times 10^{-7} \, \text{T m/A} \).

**Step 9: Substitute the values into the formula for B.**
\[ B = (4\pi \times 10^{-7}) \times (79.6) \]

**Step 10: Calculate B.**
\[ B \approx 1.0 \times 10^{-5} \, \text{T} \, \text{(approximately)} \]

### Final Answers:
- Magnetic Field Intensity (H) = 79.6 A/m
- Magnetic Induction (B) = \( 1.0 \times 10^{-5} \, \text{T} \)