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 magnetic field inside a cylindrical conductor and determine which of the provided statements is incorrect. The magnetic field inside the conductor is given by:

\[ B = B_0 r^2 \quad (0 < r \leq R) \]

where \( R \) is the radius of the cylindrical conductor and \( r \) is the distance from the central axis.

### Step 1: Calculate the Current Density \( J \)

The current density \( J \) is defined as the current per unit area. We can find the current density using Ampère's law, which states:

\[ \mu_0 I_{\text{enc}} = \oint B \cdot dl \]

For a cylindrical conductor, the current enclosed \( I_{\text{enc}} \) at a radius \( r \) can be expressed as:

\[ I_{\text{enc}} = J \cdot A = J \cdot \pi r^2 \]

Where \( A \) is the cross-sectional area of the cylinder at radius \( r \).

Using Ampère's law, we can write:

\[ \mu_0 I_{\text{enc}} = B \cdot (2 \pi r) \]

Substituting \( B = B_0 r^2 \):

\[ \mu_0 I_{\text{enc}} = (B_0 r^2) \cdot (2 \pi r) = 2 \pi B_0 r^3 \]

Now, substituting for \( I_{\text{enc}} \):

\[ \mu_0 J \cdot \pi r^2 = 2 \pi B_0 r^3 \]

Dividing both sides by \( \pi \):

\[ \mu_0 J r^2 = 2 B_0 r^3 \]

Now, solving for \( J \):

\[ J = \frac{2 B_0 r^3}{\mu_0 r^2} = \frac{2 B_0 r}{\mu_0} \]

### Step 2: Analyze the Options

1. **First Option**: The current density at a distance \( r \) from the central axis is \( J = \frac{2 B_0 r}{\mu_0} \). This is correct based on our calculation.

2. **Second Option**: We need to check if this option is incorrect. We will analyze the total current \( I_0 \) flowing through the entire conductor.

### Step 3: Total Current Calculation

Using Ampère's law for the entire conductor:

\[ \mu_0 I_0 = B_0 R^2 \cdot (2 \pi R) \]

Thus,

\[ I_0 = \frac{B_0 R^2 \cdot 2 \pi R}{\mu_0} = \frac{2 \pi B_0 R^3}{\mu_0} \]

### Step 4: Half of the Total Current

To find the radius at which half of the total current \( I_0/2 \) is enclosed, we set:

\[ \mu_0 \frac{I_0}{2} = B_0 R_0^2 \cdot (2 \pi R_0) \]

Substituting \( I_0 \):

\[ \mu_0 \cdot \frac{2 \pi B_0 R^3}{\mu_0} \cdot \frac{1}{2} = B_0 R_0^2 \cdot (2 \pi R_0) \]

This simplifies to:

\[ \pi B_0 R^3 = \pi B_0 R_0^3 \]

Thus, we find:

\[ R_0^3 = \frac{R^3}{2} \]

So,

\[ R_0 = R \cdot \left(\frac{1}{2}\right)^{1/3} \]

### Conclusion

After analyzing the options, we find that:

- The first option is correct.
- The second option is incorrect.
- The third option is also correct.
- The fourth option is correct.

Thus, the incorrect statement is the **second option**.