Consider a long cylindrical wire carrying current along the axis of wire. The current distributed non-uniformly on cross section. The magnetic field B inside the wire varies with distance `r` from the axis as `B=kr^(4)` where k is a cosntant. The current density `J` varies with distance `r` as `J=cr^(n)` then the value of `n` is (where c is another constant).

Explanation

To solve the problem, we need to find the value of \( n \) in the current density \( J = c r^n \) given that the magnetic field \( B \) inside the wire varies with distance \( r \) from the axis as \( B = k r^4 \).

### Step-by-Step Solution:

1. **Understanding the Magnetic Field**:
The magnetic field inside the wire is given by:
\[ B = k r^4 \]
where \( k \) is a constant.

2. **Applying Ampere's Law**:
According to Ampere's Law, for a cylindrical wire, we have:
\[ \oint B \cdot dl = \mu_0 I \]
where \( I \) is the current enclosed by the loop of radius \( r \).

3. **Calculating the Left Side of Ampere's Law**:
The left side can be expressed as:
\[ B \cdot 2 \pi r \]
Substituting the expression for \( B \):
\[ (k r^4) \cdot (2 \pi r) = 2 \pi k r^5 \]

4. **Calculating the Right Side of Ampere's Law**:
The current \( I \) can be expressed in terms of the current density \( J \):
\[ I = \int_0^r J \, dA \]
The differential area \( dA \) in cylindrical coordinates is:
\[ dA = 2 \pi r \, dr \]
Therefore, we have:
\[ I = \int_0^r J \cdot 2 \pi r \, dr = \int_0^r (c r^n) \cdot 2 \pi r \, dr = 2 \pi c \int_0^r r^{n+1} \, dr \]

5. **Integrating the Current Density**:
The integral can be computed as:
\[ \int_0^r r^{n+1} \, dr = \left[ \frac{r^{n+2}}{n+2} \right]_0^r = \frac{r^{n+2}}{n+2} \]
Therefore:
\[ I = 2 \pi c \cdot \frac{r^{n+2}}{n+2} \]

6. **Equating Both Sides**:
Now we equate the left side and the right side of Ampere's Law:
\[ 2 \pi k r^5 = \mu_0 \left( 2 \pi c \cdot \frac{r^{n+2}}{n+2} \right) \]

7. **Canceling Common Terms**:
We can cancel \( 2 \pi \) from both sides:
\[ k r^5 = \frac{\mu_0 c}{n+2} r^{n+2} \]

8. **Comparing Powers of \( r \)**:
Now, we can compare the powers of \( r \):
\[ 5 = n + 2 \]
Solving for \( n \):
\[ n = 5 - 2 = 3 \]

### Final Answer:
The value of \( n \) is \( 3 \).