Magnetism
Beginner
Magnetic Field of Current
Source: High school physics (Chinese)
Problem
A solenoid has length $L = 50 \text{ cm}$ and radius $r = 1 \text{ cm}$, with a total of $N = 400$ turns carrying a current $I = 3 \text{ A}$. Find the magnetic flux through the solenoid's central cross-section.
$\Phi = \mu_0 n I \pi r^2 \approx 9.47 \times 10^{-7}$ Wb $\approx 9.5 \times 10^{-7}$ Wb.
Turns per unit length: $n = \dfrac{N}{L} = \dfrac{400}{0.50} = 800 \text{ turns/m}$.
Since $L \gg r$, the field at the center is uniform:
$$B = \mu_0 n I = (4\pi \times 10^{-7})(800)(3) \approx 3.02 \times 10^{-3}\text{ T}$$Cross-sectional area: $S = \pi r^2 = \pi (0.01)^2 = \pi \times 10^{-4}\text{ m}^2$.
Flux through the central cross-section (the field is along the axis, normal to $S$):
$$\Phi = BS = \mu_0 n I \cdot \pi r^2 \approx (3.02 \times 10^{-3})(\pi \times 10^{-4}) \approx 9.47 \times 10^{-7}\text{ Wb}$$