线圈的磁场

Magnetic Field at Center of Loop

Link to image above

$$dB = \frac{\mu_o}{4\pi} \cdot \frac{I \vec{dl} \sin{90}}{r^2}$$ $$\int dB = \int \frac{\mu_oI\vec{dl}}{4\pi r^2}$$ $$\int dB = \int \frac{\mu_oI}{4\pi r^2}\int \vec{dl}$$ $$B = \frac{\mu_oI}{4\pi R^2}(2\pi r)$$ $$B = \frac{\mu_oI}{2r}$$

Magnetic Field on Loop Axis

Link to image above

$r = \sqrt{x^2+R^2}$

$\sin \theta = \frac{R}{r}$ $$dB = \frac{\mu_o}{4\pi} \cdot \frac{I \vec{dl}\sin \theta}{r^2}$$ $$dB = \frac{\mu_o(I)}{4\pi r^2}\cdot \frac{R}{r}\cdot dl$$ $$dB = \frac{\mu_o(I)(R)}{4\pi r^3}\cdot dl$$ $$dB = \frac{\mu_o(I)(R)}{4\pi (x^2 + R^2)^{3/2}} \cdot dl$$ $$\int dB = \frac{\mu_o(I)(R)}{4\pi(x^2 + R^2)^{3/2}} \int dl$$ $$B = \frac{\mu_o(I)(R)}{4\pi(x^2 + R^2)^{3/2}} (2\pi R)$$ $$B = \frac{\mu_o(I)(R)^2}{2(x^2 + R^2)^{3/2}}$$

Example 1:

What is the magnetic field at the center of a thin, conducting ring with a diameter of $0.8m$ and a current of $2.2A$?

$$B = \frac{\mu_oI}{2r}$$ $$B = \frac{(4\pi \times 10^{-7})(2.2)}{2(0.4)}$$ $$B \approx 3.456 \times 10^{-6}T$$

Example 2:

A single-turn wire loop is $2.0cm$ in diameter and carries a $650mA$ current. Find the strength of the magnetic field (a) center of the loop and (b) on the loop axis, $20cm$ from the center

(a)$$B = \frac{\mu_oI}{2r}$$ $$B = \frac{(4\pi \times 10^{-7})(0.65)}{2(0.01)}$$ $$B \approx 4.08 \times 10^{-5}T$$

(b)$$B = \frac{\mu_o(I)(R)^2}{2(x^2 + R^2)^{3/2}}$$ $$B = \frac{(4 \pi \times 10^{-7})(0.65)(0.01)^2}{2(0.20^2 + 0.01^2)^{3/2}}$$ $$B \approx 5.086 \times 10^{-9}T$$