Magnets and Earth's Magnetic Fields

Bar Magnet:

A bar magnet is a rectangular or square piece of an object made from iron or steel having permanent magnetic properties with two poles: north and south.

Bar Magnet as an Equivalent Solenoid:

 By calculating the axial field of the current carrying the final solenoid, it can be shown that the bar magnet is a solenoid. Consider a solenoid of radius a and length 2l with n turns per unit length and current I through the solenoid. Considering a small element of the solenoid of thickness dx located at a distance x from O such that OP = r.

  The magnetic field due to n rotates around the axis of the solenoid

Integrating X from -I to I to get the size of the entire field

From the above expression, it is clear that the magnetic moment of the bar magnet is equal to the magnetic moment of the solenoid.

Dipole in Uniform Magnetic Fields:

This section introduces the nature of the forces acting on a dipole placed in a uniform field and compares it with the case where the dipole is kept in an electrostatic field. As we know, when we put iron filings on a sheet of paper around a bar magnet and tape the sheet, the fillings rearrange themselves to form a certain pattern. The filling pattern of the iron here means the lines of the magnetic field produced by the magnet. These magnetic lines give us a rough idea of the magnetic field B. But many times we need to accurately determine the magnitude of the magnetic field B. To achieve this, we place a small compass needle with known magnetic moment m and moment of inertia and let it oscillate in a magnetic field.

The torque on the needle can be given as,

The magnitude of this torque is mBsinθ. Here τ is the restoring torque, and θ is the angle between the direction of the magnetic moment (m) and the direction of the magnetic field (B).

At equilibrium, we can say that,

The negative sign in the above expression mB sinθ leads to the conclusion that the restoring torque acting here acts in the opposite direction to the deflecting torque. Also, as the value of θ is very small in radians, we can approximate sin θ ≈ θ. Therefore, using this approximation, we can write

The above equation represents a simple harmonic motion and angular frequency can be represented as,

and thus, the time period can be stated as,

Or, we can also write it as,

The expression for magnetic potential energy is derived in the same manner as we derive the electrostatic potential energy as can be seen below. The magnetic potential energy Um can be given by

Magnetic Field due to a Bar Magnet:

For axial line

Consider a current loop placed in - a plane carrying current in an anticlockwise sense as seen from the positive x-axis. Due to a small current element idl shown in the figures, the magnetic field at P is given by

The angle between idl and r is 90 because it is along the axis, while r lies in the x-y plane.

The direction of dB is perpendicular r to as shown. The vector dB can be resolved into two components dB cosθ  along the z-axis and dB sinθ along the x-axis. For any two diametrically opposite current elements, the components add up while the other two components cancel out.  Therefore, the field at P is due to x- the component of the field only. Hence, we have

For an equatorial point, we assume the coil is equivalent to a magnetic dipole

 

Bar Magnet in a Uniform Magnetic Field:

When you place a bar magnet in a uniform magnetic field, the two poles of the bar magnet experience a force equal in magnitude and opposite in direction that does not have the same line of action. These forces constitute a couple that produces a turning effect called torque.

 The torque tries to rotate the magnet to align it parallel to the direction of the field. In this article, we will learn about the bar magnet, the magnetic field lines, and the bar magnet as an equivalent solenoid. We will also discuss the torque and the potential energy of a bar magnet when placed in a uniform magnetic field.

Bar Magnet

When we sprinkle some iron filings on a sheet of glass placed over a short bar magnet, these iron filings form a pattern that shows that the magnet has two poles similar to the positive and negative charge of an electric dipole; in which one pole is named as the North pole and the other as the South pole. When we suspend a bar magnet freely, these poles point approximately towards the geographic north and south poles, respectively. Around a current-carrying solenoid, a similar pattern of iron filings is also observed.

Magnetic Field Lines:

Some important properties of the magnetic field lines are given following:

The magnetic field lines form closed continuous loops, and also they do not intersect with each other.

The tangent to the field line at a given point gives the direction of the net magnetic field B at that point.

More the number of field lines crossing per unit area, the stronger is the magnitude of the magnetic field B

Torque on Magnetic Bar Placed in a Uniform Magnetic Field

The magnetic field lines give us an approximate idea of the magnetic field. To determine the magnitude of B accurately, let us consider a small compass needle of known magnetic moment (m) and moment of inertia (I). Then place this compass needle in the uniform magnetic field and allow it to oscillate, as shown in the figure below.

As we know that:

τ = Magnetic Force × perpendicular distance

Then, the torque on the needle can be expressed as:

τ = m*B

Gauss's Law of Magnetism:

 

Carl Friedrich Gauss first proposed Gauss's law in 1835, which related electric fields at points on a closed surface to the net charge contained on that surface.

The magnetic flux passing through a closed surface is governed by Gauss's law of magnetism. Here the area vector points to the surface.

Since the magnetic field lines are continuous loops, all closed surfaces have both incoming and outgoing magnetic field lines. Therefore, the net magnetic flux through the closed surface is zero.

The net current is given mathematically by the expression ϕ =B.dA, where B corresponds to the magnetic field and A represents the area.

Earth's Magnetic Field:

The Earth's magnetic field is also called the geomagnetic field. Earth's magnetic field extends millions of kilometers into outer space and looks very similar to a bar magnet. The South Magnetic Pole of the Earth is actually near the North Pole and the North Magnetic Pole is in Antarctica! Therefore, the north pole of a compass magnet actually points north (the north and south poles attract each other).

The Earth's magnetic field extends far but is very weak in terms of field strength. Only 10,000 nT compared to a refrigerator magnet with a strength of 10. The Earth's magnetic field extends far but is very weak in terms of field strength. Only 10,000 nT compared to a refrigerator magnet with a strength of 10^7 nT!.

Theory of Terrestrial Magnetism

There is one theory that explains how the Earth's magnetism occurs:

Dynamo effect:

Earth gets its magnetic field lines because it has metallic fluids that exist in both the outer and inner core. The outer core consists of molten iron, while the inner core consists of solidified elements.

What causes the Earth's magnetism?

The Earth's magnetism results from convection currents of molten iron and nickel in the Earth's core. These currents carry a stream of charged particles and generate magnetic fields. This magnetic field deflects ionizing charged particles away from the sun (called the solar wind) and prevents them from entering our atmosphere. Without this magnetic shielding, the solar wind could slowly destroy our atmosphere, preventing life on Earth. Mars lacks a strong atmosphere that could

Supports life because it lacks a magnetic field to protect it. The Earth's magnetic poles do not match the true geographic north and south poles. Instead, the magnetic South Pole is located in Canada, while the magnetic North Pole is located in Antarctica. The magnetic poles are tilted about 10 degrees relative to the Earth's axis of rotation. So the whole time the compass was pointing to Canada, not true north!

Magnetic pole

The magnetic north pole is south in northern Canada; the geographic south pole is located in the center of Antarctica, while the magnetic pole is hundreds of Kilometers away near the coast. Compasses are almost useless near the magnetic poles.

Three components are responsible for the magnitude and direction of the Earth's magnetic field:

·       Magnetic declination

·       Magnetic inclination or angle of inclination

·       The horizontal component of the earth's magnetic Field.

Magnetic declination

Magnetic declination is defined as the angle between True North and Magnetic North. In the horizontal plane, True North is never constant and constantly changes Depending on the position of the earth and time.

Magnetic tilt

Magnetic tilt is also known as tilt angle. This is the angle that the horizontal plane makes on the ground. The Magnetic equator has a tilt angle of 0°and the magnetic Poles have a tilt angle of 90°.The horizontal component of the earth's magnetic field.

 The strength of the Earth's magnetic field is explained by two components:

Horizontal component (H)

Vertical component (V)

Magnetic declination

There are three types of bottoms:

True bottom, net bottom, and magnetic bottom.

According to Nathaniel Bowditch, an American practical navigator, magnetic declination is the angle between the magnetic and geographic meridians in any place, expressed in degrees and minutes east or west, to indicate the direction of Magnetic north from true north.

Magnetic Declination

Magnetic declination is defined as the angle between magnetic north and true north in the horizontal plane, which is not constant and constantly changes with position and time on the Earth. The Greek letter δ is used as a symbol for magnetic declination and is also known as magnetic variation. If the magnetic base is east of true north, the declination is positive, and if the magnetic base is west of true north, the declination is negative. Other terms used are isogonic lines (when the lines along the declination are constant) and agonic lines (when the lines along the declination are zero).

True North

True North is defined as the direction along the Earth's Surface towards the True North Pole or the Geographic North Pole. This is also known as geodetic north and Differs from magnetic north, which is the direction of the Compass points, and grid north, which is north along gridlines.

Grid North

Grid north is defined as the direction north along the map projection gridlines. This term is used in navigation and The deviation of the north grid from the true north is much smaller.

Magnetic North

Magnetic North is defined as the direction a compass needle points in response to the Earth's magnetic field.

The deviation between true north and magnetic north is sometimes different because the Earth's magnetic poles are not fixed on its axis.

Difference between Magnetic North and True North

Here is a diagram that explains magnetic north and true north:

Magnetic base True base

This is the north direction indicated by the compass needle, which is along the earth's magnetic field. This is geographic north, which points to the North Pole.

What is a magnetic cup?

Magnetic declination is defined as the angle that the earth's magnetic field lines make with the horizontal plane. It is also known as declination or magnetic declination and was discovered by Georg Hartmann in 15.

If the declination is positive, it indicates that the

Magnetic lines of the earth point down in the northern hemisphere, and if the declination is negative, it indicates, that the earth's declination is positive. Magnetic lines

Point up in the southern hemisphere. In 1581, Robert Norman discovered the inclined circle, a method used to measure the measured angle. Other terms used are isocline lines (when the contours are equal on the ground) and inclined lines (when there are zero depressions at the location of the points).

In 1581, Robert Norman discovered the inclined circle, a method used to measure the measured angle. Other terms used are isoclinic lines (when the contours are equal on the ground) and inclined lines (when the points have zero lows).

How to calculate magnetic declination?

Here are the different ways to calculate magnetic declination:

About the Declination Calculator:

The Declination Calculator is an easy way to calculate

Declination anywhere on Earth. The calculator gives the declination based on the magnetic reference models, giving the year, latitude, and longitude of the given place.

About the Magnetic Declination Chart:

 A Magnetic Declination chart is a map that shows the Earth's available magnetic fields. About Compass: There are three types of bearings, they are true, magnetic, and compass bearings. A compass can be used to calculate the declination because it is one of the compass errors and the other is magnetic variation. These three are related:

T=M V

M =CD

T = C V D (which is the general equation relating

Compass and true bearings)

where,

C is the compass

M is the magnetic bearing

T is the actual direction

V is variation

D is the compass deviation

V<0>0,D>0for eastern variation and deviation

The following is a way to calculate the compass

Direction from the true direction:

True Bearing-Variation=Magnetic Bearing

Magnetic Bearing-Deviation=Compass Direction

Here is the way to calculate the actual direction of the compass:

Compass Bearing Deviation=Magnetic Bearing

Magnetic bearing variation=true bearing