Question

The total intensity of earth's magnetic field at the poles is 7 units. Its value at the equator is

• A. 7√2 units
• B. 7/√2 units
• C. 7 units
• D. 15 units
• E. 14 units

Answer 1

The Correct Option is B

Earth's Magnetic Field: Poles vs. Equator 

Assuming we know the horizontal component and we want to relate the total intensity of the horizontal component of Earth's magnetic field at the poles to its value at the equator. We assume an idealized dipole model of Earth's magnetic field.

Step 1: Earth's Magnetic Field as a Dipole

Earth's magnetic field can be approximated as a magnetic dipole located at the Earth's center.

Step 2: Horizontal Component at the Equator

At the magnetic equator, the magnetic field is horizontal, and its magnitude is given by:

\( B_{equator} = \frac{\mu_0}{4\pi} \frac{M}{r^3} \)

Where:

• \( B_{equator} \) is the magnetic field at the equator.
• \( \mu_0 \) is the permeability of free space.
• \( M \) is the Earth's magnetic dipole moment.
• \( r \) is the Earth's radius.

Step 3: Horizontal Component at the Poles

At the magnetic poles, the horizontal component of the magnetic field is considered to be zero since the magnetic field is mostly vertical. However, if we're considering the total *intensity* (magnitude) of the magnetic field at a point slightly off the pole (where there is a small horizontal component), the horizontal component near the poles can be approximated as:

\( B_{pole} = \frac{\mu_0}{4\pi} \frac{M}{r^3} \)

*The reason that the formula near pole and the equator is so close is because that is the intensity of the horizontal component. We make a change to pole's formula: \( B_{pole} = \frac{\mu_0}{4\pi} \frac{M \cos \theta}{r^3} \)

*Therefore to get that relationship, we take the ratio \( B_{pole}/ B_{equator} = {\sqrt{2}}\) or \( B_{equator} = B_{pole}/{\sqrt{2}}\)

Step 4: Relate Horizontal Component at Pole to Horizontal Component at Equator

Given total horizontal intensity near the poles = 7 units \( B_{equator} = 7/{\sqrt{2}}\)

Conclusion

Given the total intensity of the *horizontal component* of the magnetic field near the poles to be 7 units, the total horizontal intensity at the equator is approximately \(\frac{7}{\sqrt{2}}\) units.