Magnetic Field of a Dipole

What is the formula for the axial magnetic field at an arbitrary point?

What is the formula for the equatorial magnetic field at an arbitrary point?

How is the resultant magnetic field at point P calculated?

What is the magnitude of the magnetic field given values of m and r?

What initial horizontal and vertical velocities are given for the stone in projectile motion?

What is the horizontal distance travelled by the stone after 3 seconds?

What is the maximum height reached by the stone?

What forces act on the mass in a conical pendulum?

What is the condition for horizontal circular motion in a conical pendulum?

What is the definition of magnetic declination?

Where is the magnetic north pole of the Earth located?

What is the orientation of the magnetic equator?

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.

This quiz covers the calculations related to the magnetic field at an arbitrary point due to a magnetic dipole. It includes the analysis of axial and equatorial components, as well as examples to reinforce the concepts. Test your understanding of the formulas and applications in physics.