Author: eric2i

In general, by utilizing Ampère’s Circuital Law, it is straightforward to derive the formula for the magnetic induction intensity around a uniformly current-carrying cylindrical surface, except on the surface of the cylinder itself:

$$B(r)=\begin{cases}
0 & if 0 < r < R \\
\frac{\mu_0 I}{2\pi r} & if r > R \\
\end{cases}$$

For the case where \(r=R\), Ampère’s Circuital Law cannot compute the value.

Instead, we use the Biot-Savart Law, treating the infinitely long cylindrical surface as an infinite straight current. The surrounding field strength formula is then:

$$B(r)=\frac{\mu_0 I}{2\pi d} $$

where \(i=\frac{ds}{2\pi R}I\).

It is intuitive to place the research point on the cylindrical surface at the coordinate origin and use polar coordinates to rewrite the equation. Given that:

$$ds = \sqrt{\rho^2 + \rho^{‘2}} = 2R\theta$$

$$d=\rho$$

Thus, the differential magnetic field \(dB\) is given by:

$$dB = \frac{\mu_0 I}{4\pi^2 R}d\theta$$

Integrating from \(-\pi/2\) to \(\pi/2\), we find:

$$B = \int_{-\pi/2}^{\pi/2}\frac{\mu_0 I}{4\pi^2 R}d\theta = \frac{\mu_0 I}{4\pi R}$$