Magnetic Field of a Dipole

Magnetic Field at an Arbitrary Point

• The magnetic field (B) at an arbitrary point P due to a magnetic dipole is the vector sum of its axial ($B_a$) and equatorial ($B_e$) components, which are functions of the distance from the dipole (r) and the angle (θ) between the dipole moment (m) and the line connecting the dipole to the point.
• The magnitude of the resultant magnetic field $B$ can be calculated using the formula: $B = \sqrt{B_a^2 + B_e^2} = \frac{\mu_0 m}{4\pi r^3} \sqrt{3 \cos^2\theta + 1}$
• The angle $\alpha$ between the direction of the magnetic field $\vec{B}$ and the line connecting the dipole to the point $\vec{r}$ is calculated using the formula: $\tan\alpha = \frac{B_e}{B_a} = \frac{1}{2}(\tan\theta)$

Example 12.1

• A short magnetic dipole with a magnetic moment ($m$) of 0.5 Am² at a distance of 20 cm from the center has a magnetic field of $1.25 \times 10^{-5} ,Wb/m^2$ along the axis.
• The magnetic field of the dipole along the equatorial line is $6.25 \times 10^{-6} ,Wb/m^2$.

Earth's Magnetism

• Earth is a huge magnet with a magnetic North Pole located near the South Geographic Pole and a magnetic South Pole located near the North Geographic Pole.
• The imaginary line connecting these poles is the magnetic axis.
• An imaginary circle on the surface of the Earth, perpendicular to the magnetic axis, is the magnetic equator.
• The vertical plane passing through the magnetic axis at a point on the Earth's surface is the magnetic meridian. It defines the direction of the Earth's magnetic field at that location.
• The vertical plane passing through the geographic axis (North and South poles) is the geographic meridian.
• The angle between the magnetic meridian and the geographic meridian at a location is called magnetic declination.