# Understanding Magnetic Forces: Laws and Applications

Table of Contents

## Laws of Magnetic Force

**Understanding Magnetic Forces: Laws and Applications-** Coulomb was the first scientist who quantitatively expressed the magnetic force existing between two isolated point poles on experimental basis. It has to be inculcated into mind that magnetic poles always exist in pairs form (that’s magnetic poles are always in pairs, and a single pole does not ever exist. For example, when we talk about magnetic poles, it always means a North pole and a South pole. North pole and a South pole never exist separately). Therefore, attainment of an isolated pole is practically impossible. (In other words, even a small particle of a magnet comprises a North pole and a South pole). The concept of an isolated pole is purely theoretical or imaginary and it has nothing to do with experiment or practical. However, all poles of a long and thin magnet are considered as point poles for practical purposes. This has been illustrated in figure 4.9.

Through the application of a Torsion Balance (a device made from a very fine wire, and which can measure extremely minute forces) Coulomb discovered that a force existing between two magnetic poles (placed on a medium)

Figure 4.10

(i). Is directly proportional to their poles force

(ii). Is indirectly proportional to the square distance existing between them

(iii). Indirectly proportional to the absolute permeability of the surrounding medium.

For example, if magnetic strength of two poles is indicated by m1 and m2, and distance between these poles is “r” (as has been illustrated in the figure 4.10) and absolute permeability of the surrounding environment is “𝜇”, then force “F” existing between the magnetic poles tends to be as follows;

F m_{1}m_{2} / 𝜇_{2} or F = K m_{1}m_{2} / 𝜇_{2} or F = Km_{1}m_{2}/ 𝜇r_{2} r in vector form

Here, r is a unit vector, which indicates “r” direction.

Or F = K m_{1}m_{2} / r_{3}. r, where F and r, are vectors

In the S.I unit system, the value of constant K is as follows;

K = 1 / 4 𝜋

∴ F = m_{1} m_{2}/ 4 𝜋 𝜇_{2 }N or F = m_{1} m_{2} / 4 𝜇_{0} 𝜇_{r }r^{2 }N … in a medium

In situation of a vector

∴ F = m^{1}m^{2} / 4 𝜋 𝜇_{3 }r =m^{1} m^{2} / 4 𝜋 𝜇_{0 }r^{2} N … in air

If m_{1} and m_{2} are kept equivalent to m and r equal to one meter in the above-mentioned equation, that’s;

m_{1} = m_{2 }= m (say); r = 1 meter; F = 1 / 4 𝜋 𝜇_{0 }N

Then, m_{2} = 1 or m = 1 weber

Thus, a unit magnetic pole may be defined as follows;

A pole, which if placed at a distance of one meter from a similar and equivalent pole in a vacuum, it repels it at a force of 1 / 4 𝜋 𝜇_{0 }Newton, is called a unit magnetic pole.

## Absolute Permeability and Relative Permeability

The state of magnetism and electromagnetism depends on a specific characteristic of a medium, which is called permeability. Every medium is supposed to have the following two permeabilities;

(i). Absolute Permeability

(ii). Relative Permeability

The higher the permeability of any material, the easier to create a magnetic field in that material. The permeability is denoted by (𝜇), and it depends on the type of a material. A vacuum or free space is generally selected as a reference medium for relative permeability. Magnetizing force (H) and flux density (B) assume a great importance in any magnetic field. If an iron bar (rod) is inserted into an air core coil, its flux density will increase multiple times without causing any increase in ampere – turns. The ratio between flux density (B) and magnetizing force (H) on this iron bar (rod), is called permeability of iron. The absolute permeability of a rod material is defined as;

𝜇 = B / H henry / meter

Or B = 𝜇 H = 𝜇_{0} and 𝜇 r H wb / m^{2}

In other words, the ratio existing between B and H in any non-magnetic medium (e.g., air) other than vacuum, is called permeability. The absolute value of a vacuum’s permeability (𝜇_{0}) tends to be as follows; and it is taken as a reference;

𝜇_{0} = 4 𝜋 x 10^{-7} henry / meter or H / m

Ferromagnetic materials, (e.g., iron, steel, nickel, cobalt, and their alloys) have several hundred times higher permeability as compared to a vacuum. Normally, quantity of the relative permeability of a vacuum in free space tends to be unity, i.e., 𝜇_{r} = 1

The relative permeability of any medium other than free space tends to be 𝜇r with respect to vacuum. The ratio of the permeability of a material to that of free space is called relative permeability or the relative permeability (𝜇 r) of a material is the ratio of its absolute permeability to the permeability of a vacuum. In other words, the relative permeability of any material equals the ratio of flux density of this material to that of the flux density created in the vacuum through the same magnetizing force.

𝜇 r = B (Material) / Bo (Vacuum) … for same H

**Magnetic Force on a Current-Carrying Conductor Lying in a Magnetic Field**

Whenever a straight current-carrying conductor is placed in a magnetic field, then such a mechanical force acts on the conductor which attempts to drive it out from the field, and the direction of the current of this force is perpendicular to the direction of the field. In figure 4.11, the magnetic field created by coils A and B, having a flux density of B-Wb / m2, has been shown with a conductor xy placed at 90°. If the length of the conductor lying on this field is *ℓ *meters, and the current flowing through it is *I *amperes, then the value of mechanical force operating on this conductor, will be as follows;

Figure. 4.11

F = B I *ℓ *Newtons or

F = 𝜇 0 𝜇 r H I *ℓ *Newtons

The actual magnetic field generated as a result of coils or permanent magnets, and the field generated as a result of the flowing of current “*I”* through a conductor, have been shown in figure 4.12 (a) and the resultant field in figure 4.12 (b).

Figure 4.12 (a, b)

## Fleming’s Left Hand Rule

The direction of force generated on a current-carrying conductor placed on a magnetic field can easily be determined through “Fleming’s Left Hand Rule”. According to this rule, if we open our left hand in such a way that the thumb, first finger, and second finger of the hand are located at 90° (right angle) from each other, as has been shown in fig. 4.13, then in such a situation, the thumb reflects the force’s (or motion) direction, the first finger (or index finger) magnetic field’s direction, and the second finger reflects the direction of current flowing through the conductor.

Figure 4.13

Remember that when a conductor lies parallel to a magnetic field, no force operates on it. All electric motors work according to Fleming’s left-hand rule; therefore, this rule is also called a motor rule. If a conductor moves a distance up to d meter on a right angle or perpendicular way, in a magnetic field, then;

Work done = Force x Distance

= B I *ℓ *newtons x d meters

= BI *ℓ *d joules

= B I. A (A being area swept through in square meter)

= Փ I joules (∵ B = Փ / A or Փ = BA)

Work done (Joules) = Flux cut (Webers) x current (Amperes)

If the conductor forms an angle 0 in the field’s direction, then B can be resolved into two components. That’s B cos θ and B Sin θ. The first component (i.e., B cos θ) is located parallel to the conductor whereas the second component (B Sin θ) is located vertically to the conductor. The first component does not have any sort of impact on a conductor (i.e., no force generates as a result of this part), whereas the following force generates as a result of the other section.

F = BI *ℓ* Sin θ Newtons

### Fleming’s Left Hand Rule Calculations

**Example 1;** A conductor lying perpendicular to a magnetic field of flux density 0.5 webers / meter² is carrying a current of 80 amperes. Find the force acting on a conductor in newtons per meter run and in pounds per foot run.

**Solution;**

*f =* BI *ℓ*, newtons

= 0.5 x 80 x 1 newtons when I = 1 meter

=40 newtons per meter Ans.

1 newton = 0.225 lb and 1 meter = 3.28 ft

Thus,* f* = 0.225 x 40 x 1 / 3.28 lb per foot

= 2.74 lb per foot run. Ans

**Example 2;** The coil of a moving coil instrument is wound with 40 turns of wire. The flux density in the gap is 0.08 webers / meter² and the effective length of the coil side in the gap is 3 centimeters. Find the force in grams acting on each coil side when the current is 25 milliamperes.

**Solution;**

Each wire will contribute to the total force *f *acting on coil side.

Force acting on one wire = B I l newtons

=0.08 x 0.025 x 0.03 newtons

Force acting on coil side = 40 x 0.08 x 0.025 x 0.03 newtons

= 0.0024 newtons

= 0.0024 x 1000/ 9.81 gram

= 0.245 gram

**Example 3;** A conductor carrying a current of 75 amp is laying perpendicular to a magnetic field of intensity 0.4 Wb /m². Find the force acting on the conductor in newtons per meter run.

**Solution;**

The current in the conductor = 75 amp

The magnetic intensity = 0.4 Wb / m²

The direction of the current and magnetic intensity is perpendicular to each other, so the force acting can be given by the formula;

*f = *B. i. l Newtons, where *f *= force acting in newton, B = flux density in wb, l = length of the conductor in meters, *I*= current in amperes.

Let us take *I = *1 m, in this case. So

*f *= 0.4 x 75 x 1 = 30

So, force acting is 30 newton/ meter, Ans.

**Example 4;** The coil of a moving coil instrument is wound with 25 turns of wire. The flux density in the air gap is 0.075 Wb / m² and the effective length of the coil sides, in the air gap is, 3 cm. Find the force in grams acting on each coil side when carrying a current of 40 milliamperes.

**Solution;**

The number of turns in the coil is 25; flux density is 0.075 wb / m².

Now, the force acting per coil side = B. i. l Newton

i.e., *f *= 0.075 x 0.040 x 0.03 Newton

It is the force acting on one coil side, there are 25 turns, so the coil sides are to be considered as;

*f *= (0.075 x 0.040 x 0.03 x 25) newton

Force in gm. can be given as;

*f *= 0.075 x 0.040 x 0.03 x 25/ 9.81 gram

= 0.23 gm, Ans.

**Example 5;** A square cross-sectional magnet has a pole strength of 1 x 10 -3 webers and a cross – sectional area of 2 cm x 2 cm. Calculate the strength at a distance of 10 cm from the pole in the air.

**Solution;**

Here pole strength *m* = 1 x 10 -3 wb, and distance = 10 cm or 0.1 m.

Field strength, *H* = *m */ 4 𝜋 𝜇 0.* *𝜇 r. r² N/Wb

= 1 x 10 -3 / 4 𝜋 x 4 𝜋 x 10 -7 x 1 x (0.1)2 N/ Wb = 6338.7 N/ Wb

**Example 6;** A point pole having a strength of 0.25 x 10 -3 webers is placed in a magnetic field at a distance of 50 cm from another pole in air. The force it experiences is 0.5 newton. Determine (a) the field intensity at that point (b) the strength of another pole.

**Solution;**

Here the strength *m* of the pole = 0.25 x 10 -3 weber, distance r = 50 m, and force experienced *F *= 0.5 Newton

(a) Field strength *H *= force F / pole strength m = 0.5 / 0.25 x 10-3 N / Wb

Again, the force *F *is given by

*F *= 0.5 newton

*F *= m1 m / 4 𝜋 𝜇0. 𝜇r . r² =0.5 N

Here, *m *= 0.25 x 10 -3 Wb, r = 50 cm = 0.5 m

𝜇 r = 1 (air), 𝜇 0 = 4 𝜋 x10 -7 H/m

Subtracting the various values as given in the problem, in the above relation, we have;

0.5 = m1 x 0.25 x 10 -3 / 4π x 4π x 10 -7 x 1 x 0.5 x 0.5 N

Or, m1 = 0.5 x 16 𝜋 ² x 10 -7 x 0.25/ 0.25 x 10 -3 = 7.898 x 10 -3 Wb

## Magnetic Field of a Coil

When current flows through a wire-made coil or loop, this coil then always acts as a magnet. And the magnetic field is generated in this magnet resembles the field created in the permanent rectangular-shaped bar magnet. If a coil is made by combining a number of loops in a series, a stronger magnetic field will thus be generated. (Remember that when current is transmitted through a piece of wire by means of bending it, then this wire assumes the shape of a loop. The magnetic force lines formed around a conductor eject from one side of the loop, and enter the loop from the other side. As such, a wire-made loop, through which current flows, operates as a weak magnet, and this loop always consists of one North Pole and one south Pole. The North pole loop is created on that side, from which magnetic force lines emit from the loop or coil, and the South pole is formed on the side from which these lines enter into the loop.

Figure 4.14

If a magnetic field of any loop is desired to be strengthened, then the wire is shaped in a coil containing many loops. Thus, individual fields of all loops are combined together to form a stronger magnetic field, which spreads both inside as well as outside the coil. The magnetic lines existing in empty spaces between the coil turns, cancel each other due to being inversely related. This coil functions as a strong bar magnet, having one of its ends as the North pole, while another end as the South pole, which has been illustrated in figure 4.14. Remember that by increasing the number of turns of a coil or by means of increasing the current flowing through these turns, its field strength enhances.

If a coil is made by overlapping wire on a straight core, such type of coil is known as a solenoid coil. (Or a cylindrical coil consisting of wire having one or more than one insulation layer sheathed on it, is called a solenoid). If the current is transmitted through this coil, a magnetic field generates around it. As long as current flows through a coil or solenoid, this bar functions as a magnet. The direction (or polarity) of a magnetic field and poles generated by the coil, can be determined by the right-hand gripping rule.

## The Right-Hand Gripping Rule

According to this rule, if a coil is held in the right hand in such a way that fingers indicate the direction of the current, then the outwardly stretched thumb will reveal the North pole end. See Fig. 4.15.

Figure 4.15 (a)

## The End Rule

We know that so long as the flow of current continues through a coil or solenoid, this bar works like a magnet. When the coil is viewed from one of its ends, and the polarity of current flowing through these turns is clock-wise, then the pole going to be formed on the nearer end will be the South pole, whereas the pole on the far end will be the North Pole. If the current flows in an anti-clockwise direction, then the nearest pole will be the North pole, whereas the pole being formed on the farthest end, will be the South pole. This has been shown in figure 4.16.

Figure 4.16 – Direction of current in the coil of a solenoid

For example, if a coil has a length *ℓ *meters and the number of turns N, and an I ampere current flows through it, then the Magneto – motive force (m.m.f) of this coil tends to be IN amperes. And solenoid magnetizing force (H) can be determined with the help of the following formula;

H = IN / *ℓ *Amperes Per Meter

## The Cork–Screw Rule

The polarity or direction of a magnetic field can also be ascertained through the corkscrew rule. According to this rule, if a screw inside a cork is rotated with a right hand, then the direction of the screw’s operation will be treated as the direction of current as well, whereas the screw’s circulatory direction, indicates the direction of the magnetic field lines. As screw’s grooves are round, therefore, magnetic field’s lines are also round. (Remember that a corkscrew is a type of instrument, through which a cork can be removed from a bottle). This rule is also known as the wood screw rule. The mutual relation between current and magnetic field has been shown in figure 4.17.

Figure 4.17 – Right-hand screw rule

**Effect of Iron Core in a Coil**

If an iron-made laminated core is inserted into a coil, or if a wire is wrapped above a core and shaped into a coil, such a coil is called a solenoid. When current flows through a cylindrical type coil, a magnetic field generates surrounding it. Thus, this coil comprising the iron core functions as a permanent magnet. Remember that the magneticity of this core sustains until the flow of current continues through the coil. As soon as the flow of current disrupts, the magnetic field found around the coil is also eliminated. A solenoid built upon an iron core has been illustrated in fig. 4.18

Figure. 4.18; Solenoid with an iron core

We know that these lines of the magnetic field emit from the North pole of the coil and re-enter the coil from its South pole side’s end. In an air-core coil (a coil containing air inside instead of an iron core), magnetic lines have to counter difficulties in passing through the middle of the coil, because it is a characteristic of magnetic lines that they pass more easily through iron as compared to the air. Therefore, it tends to disperse instead of combining together while passing through an air-core coil. As such, a strong magnetic field cannot be gained as a result of the dissipation of flux.

The magnetic field power of any coil can be increased by increasing the quantity of current flowing through it or by means of increasing the number of coil turns. Moreover, a coil’s field strength can also be increased by means of inserting iron into the coil. That’s flux density of any coil can also be increased through penetration of iron in the coil, because the iron core causes minimum reluctance or obstructions in the path of magnetic force lines. Whereas, air, offers a greater reluctance than air, and as such flux density increases significantly within the coil. As a consequence of enhancement in flux density, magnetic fields concentrate or combine, and electromagnetic strength increases quite meaningfully. This has been explained in fig. 4.19.

Figure; 4.19

**Magnetic Field of a Straight Current-Carrying Conductor**

In 1819, Mr. Oersted discovered a specific relation between electricity and magnetism. As a result of this discovery, Henry and Faraday later on conducted experiments and laid the foundation of modern electrical machinery. According to the relation expressed by Mr. Oersted, whenever an electric current is caused to pass through a straight conductor, a magnetic field generates alongside the total length of this conductor, the direction or polarity of which depends on the polarity of the current. These magnetic flux lines create concentrated circles around the conductor, and these circles are located at 90° or right angle of the conductor (that’s these magnetic lines are vertically located alongside the conductor). As a result of a change in the polarity of the current, the magnetic field’s direction also changes. However, magnetic lines continue to survive in the shape of circles. In fig 4.20 (a), a magnetic field built around a current-carrying conductor has been illustrated. While in Figures (4.20- b) and (4.20- c) magnetic force lines formed as a result of changes in the current’s polarity flowing through the carrying conductor, have been illustrated.

Figure; 4.20 (a) – Magnetic field around a current-carrying conductor

Figure 4.20 (b). Magnetic lines of force around a current-carrying conductor

The point where current is subjected to transmit through a conductor, a magnetic field generates around it. This can also be explained by means of a following experiment.

When a Compass is brought closer to a current-carrying conductor, the compass sets itself automatically with the conductor at 90°. Thus, the presence of a magnetic field around the conductor is confirmed. If a conductor is passed through cardboard by making a hole in it, as has been shown in fig. 4.21, and let the current pass through the conductor, then shape and direction of the field can be determined by placing a compass at different points of the cardboard and noting its movement on these points.it is evident from this experiment that a magnetic field exists around a conductor in concentrated circles. When the direction of the current’s flow is downwards, the polarity of the field turns out to be clock-wise. And when the supply’s polarity is changed, the direction of the flow of current also changes. That’s current’s direction becomes upwards. Thus, a counter-clockwise magnetic field results. This has been depicted in fig. 4.22.

Fig. 4.21 – Experiment with exploring the field around a conductor

Fig. 4.22 – Field setup around a current-carrying conductor

The following points have been elaborated based on this experiment;

(i). When current passes through a conductor, a magnetic field generates around it.

(ii). This magnetic field is generated in the shape of concentrated circles.

(iii). This field maintains around a conductor until the flow of current continues through the conductor.

(iv). The direction of a magnetic field changes as a result of a change in the current’s direction.

(v). This magnetic field lies perpendicular to a conductor. The distinctive relation found between a magnetic field and the direction of current can be elucidated through a conductor’s right-hand rule.

Fig; 4.23 – Right-hand rule for determining the direction of the magnetic lines of force around a straight current-carrying conductor.

The following facts have also been corroborated through these experiments;

(i). If two current-carrying conductors run parallel together, having the same value of current flowing through them, however, their direction is mutually inverse, in this case, the resulting magnetic effect is practically eliminated, because of the magnetic effect of both conductors, neutralizes each other. Therefore, the fields of both these conductors repel each other.

(ii). If two current-carrying conductors run side by side in parallel, and the quantity of current flowing through them is equal, with the same direction, then the magnetic effect thus produced will be double. Because the individual magnetic effect of both conductors joins together to generate a single field.

(iii). Magnetic effect is directly proportional to the current flowing through a conductor.

## Fleming’s Right-Hand Rule

According to this law, if a right hand is stretched in such a way that the thumb, forefinger, and middle finger are spaced apart from each other at a right angle, as indicated by fig. 4.24, then the forefinger will reveal the field’s direction, the thumb will indicate conductor’s movement direction, and middle finger e.m.f direction.

Figure 4.24; Fleming’s right-hand rule for determining the direction of an induced emf

According to this rule, the direction of e.m.f produced as a result of the movement of a conductor placed in a magnetic field can be determined. Remember, that all generators work on this principle.

## Fleming’s Left–Hand Rule

According to this rule, if we open up our left hand in such a way that the thumb, forefinger, and middle finger are placed apart from one another at 90° (right angle), as has been illustrated in fig. 4.25,

in such a condition, the thumb will indicate the motion’s direction, forefinger or index finger direction of the magnetic field, while the second finger reveals the direction of flowing current in the conductor. Through this rule, the direction of current flowing through a conductor located on the magnetic field can be determined. All electric motors work under this very rule.

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