The current density `bar(J)` inside a long, solid cylindrical wire of radius `a = 12 mm` is in the direction of the central axis, and its magnitude varies linearly with radial distance r from the axis according to `J = (J_(0) r)/(a)`, where `J_(0) = (10^(5))/(4 pi) A//m^(2)`. Find the magnitude of the magnetic field at `r = (a)/(2)` in `mu T`

Explanation

To find the magnitude of the magnetic field at \( r = \frac{a}{2} \) inside a long solid cylindrical wire with a given current density, we can follow these steps:

### Step 1: Understand the Current Density
The current density \( \bar{J} \) is given by the equation:
\[ J = \frac{J_0 r}{a} \]
where \( J_0 = \frac{10^5}{4\pi} \, \text{A/m}^2 \) and \( a = 12 \, \text{mm} = 0.012 \, \text{m} \).
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