A cylindrical conductor of radius R carrying current `i` along the axis such that the magnetic field inside the conductor varies as `B=B_(0)r^(2)(0lt r leR)`, then which of the following in incorrect?

Options

A. Current density at distance 'r' from central axis of the conductor is `J(r )=(2B_(0)r)/(mu_(0))`
B. Current density at distance 'r' from central axis of the conductor is `J(r )=(3B_(0)r)/(mu_(0))`
C. Half of the total current would be confined within the radius of `(R )/((2)^((1)/(3)))` of the conductor
D. For `r gt R` magnetic field would vary as `B(r )=(B_(0)R^(3))/(r )`

Answer

To solve the problem, we need to analyze the given information about the cylindrical conductor and the magnetic field inside it. The magnetic field \( B \) is given as \( B = B_0 r^2 \) for \( 0 < r \leq R \).

### Step-by-Step Solution:

1. **Understanding the Magnetic Field**:
- The magnetic field inside the conductor varies with the radius \( r \) as \( B = B_0 r^2 \). This indicates that the magnetic field strength increases with the square of the distance from the axis of the cylinder.

2. **Finding Current Density**:
- The magnetic field \( B \) is related to the current density \( J \) through Ampere's Law. Since \( B \) depends on \( r^2 \), the current density \( J \) must also vary with \( r \).
- We can express the current density as \( J(r) = k r^n \), where \( k \) is a constant and \( n \) is a power that we need to determine.

3. **Using Ampere's Law**:
- According to Ampere's Law, for a cylindrical conductor, the integral form is given by:
\[ \oint B \cdot dl = \mu_0 I_{\text{enc}} \]
- For a circular path of radius \( r \), the left-hand side becomes \( B \cdot 2\pi r \).

4. **Calculating Enclosed Current**:
- The enclosed current \( I_{\text{enc}} \) for a radius \( r \) is given by:
\[ I_{\text{enc}} = \int_0^r J(r') \cdot dA = \int_0^r J(r') \cdot 2\pi r' dr' \]
- Substituting \( J(r') = k r'^n \):
\[ I_{\text{enc}} = 2\pi k \int_0^r r'^{n+1} dr' = 2\pi k \left[ \frac{r^{n+2}}{n+2} \right]_0^r = \frac{2\pi k r^{n+2}}{n+2} \]

5. **Equating Both Sides**:
- From Ampere's Law:
\[ B \cdot 2\pi r = \mu_0 I_{\text{enc}} \]
- Substituting \( B = B_0 r^2 \) and \( I_{\text{enc}} \):
\[ (B_0 r^2) \cdot 2\pi r = \mu_0 \left( \frac{2\pi k r^{n+2}}{n+2} \right) \]
- Simplifying gives:
\[ 2\pi B_0 r^3 = \frac{2\pi \mu_0 k r^{n+2}}{n+2} \]
- Canceling \( 2\pi \) and rearranging:
\[ B_0 r^3 = \frac{\mu_0 k r^{n+2}}{n+2} \]

6. **Comparing Powers**:
- For the powers of \( r \) to match, we have:
\[ 3 = n + 2 \implies n = 1 \]
- Substituting \( n = 1 \) into the equation gives:
\[ B_0 = \frac{\mu_0 k}{3} \]

7. **Finding Current Density**:
- Thus, the current density becomes:
\[ J(r) = k r = 3B_0 r / \mu_0 \]

8. **Analyzing the Options**:
- Now we need to determine which of the provided options is incorrect based on our findings about \( J \) and the behavior of \( B \).

### Conclusion:
After analyzing the behavior of the magnetic field and the current density, we can conclude that:

- **Option A**: Correct
- **Option B**: Correct
- **Option C**: Correct
- **Option D**: Incorrect (as it does not follow the derived relationship for \( B \) when \( r > R \))