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Questions and Answers
What is the formula for the axial magnetic field at an arbitrary point?
B_a = \frac{\mu_0}{4\pi} \frac{2m \cos\theta}{r^3}
What is the formula for the equatorial magnetic field at an arbitrary point?
B_e = \frac{\mu_0}{4\pi} \frac{m \sin\theta}{r^3}
How is the resultant magnetic field at point P calculated?
B = \frac{\mu_0 m}{4\pi r^3} \sqrt{3 \cos^2\theta + 1}
What is the magnitude of the magnetic field given values of m and r?
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What initial horizontal and vertical velocities are given for the stone in projectile motion?
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What is the horizontal distance travelled by the stone after 3 seconds?
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What is the maximum height reached by the stone?
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What forces act on the mass in a conical pendulum?
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What is the condition for horizontal circular motion in a conical pendulum?
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What is the definition of magnetic declination?
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Where is the magnetic north pole of the Earth located?
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What is the orientation of the magnetic equator?
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Study Notes
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.
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Description
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.
