A long thin hollow metallic cylinder of radius `'R'` has a current `i` ampere. The magnetic induction `'B'` -away from the axis at a distance `r` from the axis varies as shown in

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Answer

To solve the problem of finding the magnetic induction \( B \) at a distance \( r \) from the axis of a long thin hollow metallic cylinder carrying a current \( i \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a long thin hollow metallic cylinder with radius \( R \) carrying a current \( i \).
- We need to determine the magnetic induction \( B \) at a distance \( r \) from the axis of the cylinder.

2. **Applying Ampere's Circuital Law**:
- According to Ampere's Circuital Law, the line integral of the magnetic field \( B \) around a closed loop is equal to the permeability of free space \( \mu_0 \) times the current \( I \) enclosed by that loop.
- Mathematically, this is expressed as:
\[ \oint B \cdot dl = \mu_0 I_{\text{enc}} \]

3. **Finding the Magnetic Field Inside the Cylinder**:
- For a point inside the hollow cylinder (where \( r < R \)), the enclosed current \( I_{\text{enc}} = 0 \) because there is no current flowing through the area enclosed by the loop.
- Thus, applying Ampere's Law:
\[ \oint B \cdot dl = 0 \implies B = 0 \quad \text{for } r < R \]

4. **Finding the Magnetic Field Outside the Cylinder**:
- For a point outside the hollow cylinder (where \( r > R \)), the entire current \( i \) is enclosed by the loop.
- The magnetic field \( B \) at a distance \( r \) from the axis can be calculated as:
\[ B \cdot (2\pi r) = \mu_0 i \]
- Rearranging gives:
\[ B = \frac{\mu_0 i}{2\pi r} \quad \text{for } r > R \]

5. **Summarizing the Results**:
- The magnetic field \( B \) is:
- \( B = 0 \) for \( r < R \)
- \( B = \frac{\mu_0 i}{2\pi r} \) for \( r > R \)

6. **Graphical Representation**:
- The magnetic field \( B \) is zero inside the cylinder and decreases inversely with distance \( r \) outside the cylinder.
- Thus, the graph of \( B \) versus \( r \) will show \( B = 0 \) for \( r < R \) and a hyperbolic decrease for \( r > R \).