Axial Field of Bar Magnet

Using magnetic field of a dipole on its axis for $$z \gg R, h$$: $$B = \frac{\mu_0}{4\pi} \frac{2M}{z^3}$$

Using element of a hollow cone at distance $$x$$ along the slant height $$L = \sqrt{R^2+h^2}$$:

$$r = x\sin\alpha = x\frac{R}{L}$$

$$dA = 2\pi r dx = 2\pi x \frac{R}{L} dx$$

Using surface charge density: $$\sigma = \frac{Q}{\pi R L}$$

$$dq = \sigma dA = \frac{Q}{\pi R L} \left(2\pi x \frac{R}{L} dx\right) = \frac{2Q}{L^2} x dx$$

Using magnetic moment of a rotating ring element:

$$dM = dI \cdot \pi r^2 = \left(\frac{\omega}{2\pi} dq\right) \pi r^2 = \frac{1}{2} \omega r^2 dq$$

$$dM = \frac{1}{2} \omega \left(x\frac{R}{L}\right)^2 \left(\frac{2Q}{L^2} x dx\right) = \frac{QR^2\omega}{L^4} x^3 dx$$

$$M = \int_0^L \frac{QR^2\omega}{L^4} x^3 dx = \frac{QR^2\omega}{L^4} \left[\frac{L^4}{4}\right] = \frac{1}{4}QR^2\omega$$

$$B = \frac{\mu_0}{4\pi} \frac{2\left(\frac{1}{4}QR^2\omega\right)}{z^3} = \frac{0.5\mu_0}{4\pi} \frac{QR^2\omega}{z^3}$$