Electromagnetic Devices: A Study of Fields, Laws, and Electromagnets

This report delves into the fundamentals of electromagnetic devices, commencing with an explanation of scalar and vector fields, illustrating their differences and providing examples. It then proceeds to detail Coulomb's law, which quantifies the electrostatic force between charged particles, and Gauss's law, which relates electric flux to enclosed charge. The report further examines the properties and applications of permanent magnets and electromagnets, highlighting their distinctions and advantages. The content includes detailed explanations, formulas, and examples to clarify the concepts of electromagnetism, making it a comprehensive resource for understanding the behavior of electric and magnetic fields.

Electromagnetic Devices

Table of Contents
1 Scalar and vector fields.................................................................................................................1
2 Coulomb's law...............................................................................................................................3
3 Gauss law......................................................................................................................................5
4. PERMANENT MAGNET AND ELECTROMAGNET.............................................................8
5. BIOT-SAVART LAW...............................................................................................................11
6. AMPERE'S FORCE LAW........................................................................................................14
7. TOROIDAL COIL.....................................................................................................................16
REFERENCES..............................................................................................................................20

1 SCALAR AND VECTOR FIELDS
Scalar field refers to a quantity which has only magnitude. No direction is defined to
scalar quantities. Scalar fields are addressed to be independent of coordinate system and same
units will be observed from every direction. It also describes potential energy of physical
quantities defined with a specific force. Examples of scalar field which is used are temperature,
pressure, higgs field. Scalar field is applied to many theories in physics which are illustrated
below:
· A scalar field combined with spin-0 particles in quantum field theory. Charged particles
are represented by scalar fields which is complex.
· Scalar field is used to solve horizon problem. These are supposed to cause accelerated
expansion of universe.
· Scalar field is used to illustrate gravitation field.
Vector field refers to a quantity which has moth magnitude and direction. It is dependent
on coordinate system and can be represented by arrows which shows both magnitude and
direction along with quantity.
A force is represented as a vector quantity because a force is applied in a specific direction. It
can be changed from one point to another point. Example of vector field are illustrated below:
· Air movements can be represented with vector field on earth wind has both speed and
direction and can be visualised with an arrow.
1

· A fluid which is moving is visualised as a vector quantity because it is associated with
velocity that is a vector quantity.
Difference between scalar and vector fields are described below:
Scalar field Vector field
It has only magnitude. It has both magnitude and direction.
It is one dimensional quantity. It is multidimensional quantity.
It has simple comparison between two
quantities.
It has complex comparison between two
quantities.
Operations can be performed easily in scalar
representation.
Operations can not be performed in vector
quantities.
Vector fields are most complex one. Direction is represented by an arrow (Cooray,
Rachidi and Rubinstein, 2017). In case of 3-D dimension, three unit vector are noted:
· î is represented as unit vector in x- direction.
· Ĵ is represented as unit vector in y- direction.
· K is represented as unit vector in z- direction.
Length of arrow above quantity do not represent the size or length. It is a general representation
of vector quantities.
Lets explain scalar and vector field by taking an example:
· a player who is playing football is running 20 miles per hour from starting to end. Is it
scalar or vector?
Solution: this is a vector field because it has magnitude of 20miles per hour and direction
is given from starting to ending.
· Temperature of room is observed to be 35 degree Celsius. Is it scalar or vector?
Solution: this is a scalar because it has only magnitude. Temperature do not have
direction.
· A force which is applied on a body which is shown by a point on Cartesian plane and it
has coordinates as (3,4). how can it is represented as vector form?
Solution: a point (3,4)can be represented as f(x,y) = 3î + 4Ĵ .
in this î and Ĵ are vector units which shows direction in x-plane and y- plane
respectively.
2

2 COULOMB'S LAW
Coulomb's law is a law of physics which quantifies the amount of force either between
two stationary or electrically charged particles. A electrostatic force between two charged
electrical bodies is called coulomb force. It is very important in theory of electromagnetism to
discuss quantity of electric charge. It generally gives an idea about force between two point
charges.
Therefore, Coulomb law states that “magnitude of electrostatic force of attraction or
force of repulsion between two charge points is directly proportional to product of magnitude of
charges and inversely proportional to distance between two point charges”.
Force of attraction or repulsion on the basis of above statement can be represented as
F q1*q2∝
F 1/ r∝
F q1*q2/r∝
hence it can be formulated as to calculate force of attraction or repulsion is
F = k q1*q2/r
here, F = force of attraction
q1, q2 = quantity of charge points
r = distance between two charge points
k = proportionality constant = ¼ π ε₀ = 8.987551787 * 10 N. m.C˜
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There are three conditions which needs to be assessed while considering coulomb's law:
· charge should have spherical distribution symmetrically. It can be either point charges or
charged metal sphere.
· Overlapping of charges should not be there.
· Charges must be stationary with respect to each other.
According to this law, charges with opposite signs attract each other and charges with same sign
repel each other. Size of charges is small as compared to distance between toe charges. Therefore
size of charge is approximated as negligible (Halliday, Resnick and Walker, 2016). SI unit of
electrostatic force is newton, distance is in metre and charges is in coulomb.
Let's take examples to calculate coulomb force between two point charges.
· Two charges of value 1C and -9C are kept at a distance of 9m. Find coulomb force and
show that it is attraction force or repulsion force.
Solution : coulomb force can be calculated by formulae F = k q1*q2/r
here, F = 9*10 * 1*9/ 9
F = 10 newton.
As charges are of opposite sign,
it represents that it is an electrostatic force of attraction.
· Two point charges has a value of +3.37 μC and -8.21 μC. Force of attraction is given as
0.0626 newton. Find the distance between two point charges.
Solution: coulomb force can be calculated by formulae F = k q1*q2/r but here, distance needs
to be calculated. So, by putting values in above formulae:
0.0626 = 9*10 * 3.37* 8.21/ r
= r = 9*10 * 3.37* 8.21/ 0.0626
= r = sqrt (3.98)
= r = 1.99 m
Coulomb's law to find electrostatic force can be compared with gravitational force
because their point charges are replaced by mass of body. Distance between bodies or charges is
inversely proportional to gravitational force or electrostatic force respectively.
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3 GAUSS LAW
Gauss law deals with study of electric charge and electric flux. It is applicable for closed
surface which has definite volume. Electric flux is defined as number of electric lines that goes
through a virtual surface. It can be stated as rate of flow of electric field across an area or
surface.
Gauss law states that “electric flux across a closed surface is directly proportional to total
electric charge enclosed under surface or area”. It is included in Maxwell's law of
electromagnetism. A positive electric charge is represented to generate positive electric field
only. And similarly with negative charge which generate negative electric field. Negative charge
will act as a sink while positive charge act as a source (Gaete, Helayël-Neto and Ospedal, 2019).
An integral equation is adopted to explain gauss law:
∫E⋅dA=Q/ε0
here,
E = electric field vector
Q = enclosed electric charge
ε0 = electric permittivity
A = pointed normal vector
5

Using gauss law, flux can be measured by analysing electric field enclosed in an area,
Φ=∫E⋅dA
Since, electric flux, Φ = Q/ε0
It can be stated according to gauss law that net electric flux across a closed area is zero.
But there is a condition involved in this that it will be implied unless volume bounded by closed
surface contains no net charge.
Lets take an example to understand the concept of electric flux according to gauss law:
· Three charges of 6C, 3C and 5C are enclosed in an area. Calculate total flux that can be
generated.
Solution: total charge can be calculated by addition of different charges.
1. Q = 6C + 3C + 5C
2. Q = 14C
Total flux ϕ = Q/ϵ0
ϕ = 14/(8.854 * 10¯)
ϕ =1.584 Nm/C
Gauss law can be formulated as similar to Gauss- inverse square law. Because gauss law is also
inversely proportional to distance between point charges.
Hence, E = Q/ (4 π ε₀ r)
E is electric field which is a vector quantity. There are two conditions at which this works. Let's
take a sphere of radius R which has distributed charge Q. electric field can be outward or inward
as pointed from centre. R is distance between centre of sphere and electric field.
ΦE = E 4 π r = Qinside/ϵ0 = Q/ϵ0
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It is a derived equation according to gauss law and therefore, electric field is calculated as
E = Q/ (4 π ε₀ r)
1.
In this condition, r is greater than R. here entire charge is enclosed by surface. Field is outside of
sphere and distributed spherically and symmetrically.
· If Q is positive then the field directs outwards.
· If Q is negative then the field directs inwards.
Qinside can be represented as charge density where ρ = Q/V
Here volume of sphere Q = 4πR³/3
2.
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From this, electric field can be calculated in this condition through E = (ρR³/3ε₀r)n. Here, n
showing directional vector quantity of electric field.
In this condition, r is smaller than R. electric field is inside sphere. But also distributed
spherically an symmetrically. Here to calculate electric field, distance is found to be negligible,
therefore, r is eliminated. Hence, formulae can be expressed as
E = (ρR³/3ε₀)n
4. PERMANENT MAGNET AND ELECTROMAGNET
Permanent Magnet
The property of the permanent magnet is it can retain its magnetism and magnetic
property for long time. The permanent magnet is a material or an object which produces the
magnetic field. The permanent magnet is made of the alloy of aluminum, nickel and cobalt.
Other type of the permanent magnet are Neodymium, Samarium Cobalt, Ceramic(Ferrite). There
are two types of permanent magnets natural magnet and other one is artificial magnet (Wu and
El-Refaie, 2018). The example of the artificial magnet is Alnico and the example of the natural
magnet is Samarium Cobalt magnet. The bar magnet is shown in the picture is the best example
of permanent magnet and the lines around the magnet is the representation of the magnetic field.
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Electromagnet
The electromagnet is a solenoid which is wrapped with coil which is connected to the
electric DC battery. If the circuit of this coil is open then there will no change occur in the set up
but when the circuit is closed and current is flowing through coil the solenoid start behaving like
magnet and starts to produce the magnetic field. This magnetic property remains in the solenoid
till the circuit is closed. This solenoid looses its magnetic property as the circuit opens. This
solenoid can be any ferromagnetic material (Farjoud and Bagherpour, 2016).
BL=μNI
B=μ(N/L)I
B=μnI
B= Magnetic Field
N= Total number of turns
I= Current in the coil
n= Number of turns in L Length
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Difference Between Permanent and Electromagnet
Permanent Magnet Electromagnet
· The magnetic property of this type of
magnet is permanent.
· The magnetic property is depended on
the electrical current supply.
· These are usually made of hard material
like Nickle cobalt.
· These are normally made of soft
materials like iron.
· The magnetic field strength is constant
and permanent.
· The strength of the electromagnet id
dependent on the flow of electrical
current.
· The poles of the permanent magnet are
constant and can not be changed.
· The poles can be change in the
electromagnet by changing the direction
of current in the coil.
The property of the magnetic field are same both are consists of north and south poles
and their behavior depends on the geographical north and south pole of the earth. The property of
the magnetic field in both permanent magnet and electromagnet are same the only difference is
electromagnet is based on electric current.
Advantages of Permanent Magnet
1. There is no need of electrical current in permanent magnet.
2. There is not any problem with electrical contact.
3. Cost efficient
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Disadvantages of permanent Magnet
1. This type of magnets can't be used in the hot temperature. Higher temperature can
damage the magnetic property of the permanent magnet.
2. Poles of the permanent magnet are constant.
3. The permanent magnet tends to corrode with the time.
4. Strength of permanent magnet reduces with the time.
Advantages of Electromagnet
1. Provide the control over magnetic field.
2. Magnetic strength can be controlled.
3. Magnetic Power provided by the electromagnet is higher.
Disadvantages of Electromagnet
1. needs large number of copper coils
2. For higher magnetic power large coil is required which increases its size which not good
for functionality.
3. Requires maintenance time to time.
4. There is danger of short circuit in the electromagnet.
5. Needs continuous current supply.
6. The field of electromagnet is affected by Ohmic heating and voltage spikes and core loss.
5. BIOT-SAVART LAW
“The intensity of the magnetic field due to a constant current in the infinite long
conductor or a straight wire is directly proportional to the current and inversely proportional to
the distance of point from the wire”.
Both the electric field and magnetic field seems different but in combine way they are
known as electromagnetic force. When a charge carrying particle is not moving it generates the
electric field and when it is moving it also generates the magnetic field. Current flows through a
wire also produces the magnetic field around the wire and it can be evaluated by a compass. The
needle of compass deflects as it puts near to the current carrying conductor. Bio t Savart law is a
equation that explains the magnetic field produced by a current carrying conductor and allow
calculating the strength of magnetic field at different points (Kimura and Moffatt, 2017).
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Biot Savart law is used to determine the intensity of the magnetic (H) near a current carrying
conductor. The direction of the magnetic field follows the Right Hand rule.
The magnetic flux density dB is directly proportional to the length of the conductor dl,
current I and Sine of Ɵ which is angle between the direction of current and the vector join the
given point. The magnetic field strength is inversely proportional to the square of destance.
The equation from the Biot Savart Law
dB (∝ Idlsin Ɵ)/r
dB=k(idlsinƟ )/r
here k=(μ₀μr )/4Π
dB= (μ₀μr )/4Π*(Idlsin Ɵ)/r
let us consider a long current carrying conductor and qa point P in space. Initially for the
dl length of wire which is at r distance from point P. This vector connecting point P and wire dl
with angle θ. The current in the conductor is I. The current carrying by the dl wire is equal tu the
current in the whole wire.
So,
dB I∝
the magnetic force intensity is inversely proportional to the square of distance r.
dB 1/r∝ 2
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The magnetic field density is directly proportional to the length of wire dl and at angle of θ. The
distance between the P and dl is r.
Hence dB dlsinθ∝
By combining the above statement we can write this equation-
dB (I.dl.sinθ)/r∝ 2
this is the basic equation of the Biot Savart law. Whem the value of the constant K is puted in the
equation will be.
dB = k*(I.dl.sinθ)/r2
here value of k,
k=(μ₀μr )/4Π
the μ₀ used in the expression is absolute permeability of air and its value is 4Π10-7 Wb /A-m in
the SI system of units. And μr is the relative permeability of the medium.
The flux density at point P due to the total length of the current carrying wire can be given as
B = ∫dB = ∫μ0 μr/4п x Idl Sin θ/ r2 = I μ0 μr/4п ∫ Sin θ/ r2 dl
The D is the perpendicular distance of point P from the wire. Then
r Sin θ = D => r = D/ Sin θ
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the expression for the magnetic flux density at point P can be given as
B = I μ0 μr/4п ∫ Sin θ/ r2 dl = I μ0 μr/4п ∫ Sin3θ/ D2 dl
Again, Cotθ = l/D then, l = Dcotθ
there for the figure above. dl- -Dcsc2θdθ
finally, the expression of B comes as,
The angle θ depends on the length of wire and the position of the point P
B = I μ0 μr/4п ∫ Sin3θ/ D2 (-D csc2θ dθ)
B = -I μ0 μr/4пD ∫ Sin3θ csc2θ dθ => –I μ0 μr/4пD ∫ Sinθ dθ
lets take the wire is infinity long, θ will very very small from 0 to Pi that is θ1= 0 and θ2=Pi by
putting these values in the final expression of Biot Savart law,
we get,
B = I μ0 μr/4пD [Cos] = I μ0 μr/4пD = I μ0 μr/2пD
and this is the equation of Ampere law.
6. AMPERE'S FORCE LAW
The full version of the Ampere's Law is one of the equation of the Maxwell's equations of
electromagnetic force. As per the Ampere law. The magnetic field generated from the current
carrying conductor is directly proportional to the size of electrical current with constant of
proportionally equal to the permeability of the free space. Stationary charge produces the
electrical field which is proportional to the magnitude of the charge carrier. The moving charge
produces the magnetic field which is directly proportional to the size of the current. The
differential equation of the Ampere law is the only problem (Duhem and Aversa, 2018). Some
calculation in the calculus is needed for this equation. To get the equation for the particular
situation some information is needed to plug in for the particular condition and complete
integral.
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This Ampere's force law for the magneto statics, the force of attraction and repulsion
between two current carrying wire is known as Ampere's Force law. The both current carrying
conductors are carrying current and if a conductor is carrying the current then due to the mobility
of the charge individual magnetic field is generated. The nature of the force is dependent on the
direction of the current. If the direction of the current is same then the nature of the force will be
repulsion. For the condition when the both conductor are carrying the current in the opposite
direction then the nature of the force will be attraction.
If two long, parallel conductors and carrying the current which is I1 and I2 . Where the ka
is the magnetic force constant, r is the separation of the wires, and I1 and I2 are the direct current
respective in conductor 1 and conductor 2. The distance between the wire is smaller compared to
the length of the conductors. The distance between the conductor is greater than the diameter of
the conductors. The field created by a long straight current carrying conductor are in form of the
concentric circles. As the distance is increases the distance between the field circles. By using
the sum the magnetic field element that produces the concentric circles which is B times of the
element delta l is equal to the mu-0 times of the current in the conductor.
Ampere's circuit law states that the line integral of the magnetic field H (the circulation of
H) around the closed path is the current enclosed by this path.
H . dl=H.l= I 2pr 2pr=I∮
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The total current is
∮H . dl=I
Ienc= ∮→j.ds
By applying stoke theorem,
H.dl=∮ s∫ *H.ds∇
s *H.ds=∮∇ s∮→j.d→s
*∇ →H=→J
7. TOROIDAL COIL
The toroid is a coil of insulated or enameled wire wound on the donut shape which is
made of the powered iron. This is used as a inductor in the electrical and electronic circuit. These
toroids are specially used for the low frequency circuit where the high amount of inductance is
necessary.
The toroid has more inductance, for the given number of turns, than the solenoid with
core of same material and similar size. These make it possible to design high inductance coil
with in a small size and mass. The coil of the toroid can carry more current than the sollenoid
because the wires with the large diameter can be used in toroids which reduces the overall
resistance. In the toroid the, all the flux is contained with in the core material. Because the toroid
has no ends which helps to prevent the flux leakage. The confinement of the flux help to prevent
the external magnetic fields from affecting the behavior of toroid. No flux leakage in the circuit
prevent the other element of the circuit from damaging. The magnetic field inside the toroid can
be given as-
B =(μ₀NI/2Πr)
in this I is the current flowing through the solenoid which is endless. r is the average radius of the
toroid and n is the number of turns per unit length. Nis equal to the 2Πrn which is equal to
perimeter of the toroid*number of turns per unit length.
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1. Magnetic field at a point P outside the toroid
at a distance r. by considering a Ampere's loop passing through P. by applying the Ampere Law-
∮ B.→dl→ = μ0i
we can observe the net current is zero because the outside there is no current.
B.dl = 0∴
B = 0
Hance, magnetic field outside the toroid is zero which helps to prevent the leakage of current.
2. At the point inside the toroid
by considering the Ampere loop passing through P
∮ B.→dl→ = μ0i
the current through the loop is again zero
because B= 0
3. magnetic field at point within the toroid
Consider an Ampere loop passing through P which has center same as that of
toroid. The magnetic field B is along the Ampere loop and it is constant. The θ is
zero.
17

B.dl = B dl cosθ = B2πr∮ ∮
Current passing through the loop is
iTotal = nl×i
iTotal = n2πr×i
now applying the Ampere law
B2πr = μ0n2πri
B = μ0ni
this represents that the overall magnetic field exist with in the toroid coil. This
property makes the toroid coil much better than other type of inductors.
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REFERENCES
Books and Journals
Cooray, V., Rachidi, F. and Rubinstein, M., 2017. Formulation of the field-to-transmission line
coupling equations in terms of scalar and vector potentials. IEEE Transactions on
Electromagnetic Compatibility. 59(5). pp.1586-1591.
Duhem, P.M.M. and Aversa, A., 2018. Ampère’s Force Law.
Farjoud, A. and Bagherpour, E.A., 2016. Electromagnet design for magneto-rheological
devices. Journal of Intelligent Material Systems and Structures. 27(1). pp.51-70.
Gaete, P., Helayël-Neto, J.A. and Ospedal, L.P.R., 2019. Coulomb's law modification driven by
a logarithmic electrodynamics. EPL (Europhysics Letters). 125(5). p.51001.
Halliday, D., Resnick, R. and Walker, J., 2016. Principles of Physics Extended, International
Student Version. India: Wiley India Pvt. Ltd.
Kimura, Y. and Moffatt, H.K., 2017. Scaling properties towards vortex reconnection under Biot–
Savart evolution. Fluid Dynamics Research. 50(1). p.011409.
Wu, F. and El-Refaie, A.M., 2018. Permanent magnet vernier machine: a review. IET Electric
Power Applications. 13(2). pp.127-137.
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