Question

Is there a magnetic field inside a conductor?

Answer 1

Step 1: Diagram

Draw the required diagram,

Step 2: Explanation

1. Ampere's Law can be used to calculate the magnetic field within a conductor with a uniform current density $J=\frac{I}{\pi R^2}$.
2. Where R is radius of conductor, ${\mathrm{r}}^{\text{'}}\mathrm{is}\mathrm{outer}\mathrm{radius},\mathrm{r}\mathrm{is}\mathrm{inner}\mathrm{radius}$
3. The magnetic field $B$ inside the conductor increases linearly with $r$.
4. The magnetic field outside the conductor becomes that of a straight conductor and diminishes with radius.

Step 3: Derivation

Magnetic field inside a conductor,

Ampere's law at $r$ gives,

$$\begin{gathered} B\times 2\pi r=\mu J\pi r^2 \\ B=\frac{\mu Jr}{2} \\ \Rightarrow B=\frac{\mu rI}{2\pi R^2} \end{gathered}$$

Which approaches the surface,

$B_{Surface}=\frac{\mu I}{2\pi R}$

Where, $I=Current,Rorr=Radius\&\mu =Permeability$.

Outside of the surface,

$B=\frac{{\mu }_{o}I}{2\pi r\text{'}}$

Hence, there is a magnetic field present inside a conductor.