Unit 4
ELECTROMAGNETISM
Q1) A light bulb emits 5.00 W of power as visible light. What are the average electric and magnetic fields from the light at a distance of 3.0 m?
Solution: Assume the bulb’s power output P is distributed uniformly over a sphere of radius 3.0 m to calculate the intensity, and from it, the electric field.The power radiated as visible light is then
Q2) The beam from a small laboratory laser typically has an intensity of about 1.0 x 10-3 W/m2. Assuming that the beam is composed of plane waves; calculate the amplitudes of the electric and magnetic fields in the beam.
Solution: The intensity of the laser beam is
I = 
c
The amplitude of electric field is therefore
=0.87 V/m
The amplitude of magnetic field can be obtained by
B0 = E0/ c
=2.9 x 10-9 T
Q3) Solar Constant At the upper surface of the Earth’s atmosphere, the time-averaged magnitude of the Pointing vector, <S> = 1.35x103 W/m2, is referred to as the solar constant.
(a) Assuming that the Sun’s electromagnetic radiation is a plane sinusoidal wave, what are the magnitudes of the electric and magnetic fields?
(b)What is the total time-averaged power radiated by the Sun? The mean Sun-Earth distance is R= 1.50x1011m
Solution: (a)The time-averaged Pointing vector is related to the amplitude of the electric field by
<S> = = 
The amplitude of electric field is therefore
The amplitude of magnetic field can be obtained by
B0 = E0/ c
= 
= 3.4 x 10-6T
The associated magnetic field is one tenth of the earth’s magnetic field.
(b) The total time averaged power radiated by the Sun at the distance R is
<P> =<S>A =<S> 4πR2
=1.35x1034 x 3.14 x (1.50x1011)2
= 3.8 x 1026 W
The type of wave discussed in the example above is a spherical wave, which originates from a “point-like” source. The intensity at a distance r from the source is which decreases is .
I =<S> = <P> /4πR2
Which decreases as 1/r2, On the other hand, the intensity of a plane wave remains constant and there is no spreading in its energy.
Q4) Write a note on displacement current?
Solution: Displacement current is the rate of change of electric displacement field.
In electromagnetism, displacement current is a quantity appearing in Maxwell's equations that is defined in terms of the rate of change of electric displacement field.
Displacement current has the units of electric current density, and it has an associated magnetic field just as actual currents do.
It is mathematically represented as
ID = JD S = S 
Where
S = Area of the plate of the capacitor
ID =Displacement current
JD =Displacement current density
D = εE where ε is permittivity of the medium
During the charging and discharging process of the capacitor, the electric current flows through the wires of the circuit. However, no current flow between the plates of the capacitor.
According to Ampere's law, the magnetic field should not present between the plates as there is no current, but in reality, the magnetic field exists there. Maxwell formulated this limitation of Ampere's law by adding a term in the equation of Ampere's law to solve the issue.
Maxwell predicted that the magnetic field will still exist even in the absence of conduction current, and the magnetic field may be associated with the changing electric field. This theory of Maxwell was experimentally proved.
Since the magnetic field is associated with the electric field, the general displacement current formula is given by,
= μ0( I+ε0
)
This equation is the generalized formula of Maxwell-Ampere law.
Displacement Current Definition
The displacement current (ID) is the part which Maxwell has added to the Ampere's law.
ID = ε0
ε0= Permittivity of free space
Derivative of electric flux
Electric flux is the time rate change of flow of the electric field through a surface. If we take the derivative of electric flux, we get the rate of change of the electric field of a given area concerning time.
Q5) Discuss the energy in an electromagnetic field?
Solution: A wave’s energy is proportional to its amplitude squared. This is true for waves on guitar strings, for water waves, and for sound waves. In electromagnetic waves, the amplitude is the maximum field strength of the electric and magnetic fields. So the energy of electromagnetic field is proportional to its amplitude squared (E2 or B2).
The electric field describes an electromagnetic wave completely in free space. The magnetic field is related to the electric field by a simple relationship. Start from Faraday's law.
∇ x E= -dB/dt
From left side . Substitute the one dimensional wave equation for electricity and find its curl.
From right side, Substitute the one dimensional wave equation for magnetism and find its time derivative.
Set the two sides equal. Cancel the cosine terms and some other stuff. We will get
E0 = fB0
After Rearranging it
=fλ
And we know thatfλ is the speed of light. So the above equation can be written as
= c
This knowledge can then be used to simplify the energy density situation a bit. Start with the magnetic energy density and replace it with an expression containing the electric field.
Recall that the speed of light is related to the permeability and permittivity constants.
c = 
So
= μ0ε0
Thus
ηB = 
μ0ε0 
= 
It's the electric energy density. For an electromagnetic wave in free space, half the energy is in the electric field and half is in the magnetic field
η = ηE + ηB
η = 
+
This gives us this compact equation for the total energy density of an electromagnetic wave.
η = ε0E2
or this one, if you prefer to state things in terms of the magnetic field instead
η = 
B2
This is an interesting and simple set of relations, but keeps in mind that it only works for electromagnetic waves in free space. Things are different in a media and the electric and magnetic fields can have any values they want if they're static (meaning there's no accelerating charges).
Since waves are spread out in space and time, energy density is often a more useful concept than energy.
Q6) For a long solenoid the magnetic field strength within the solenoid is given by the equation B=5.0t T, where t is time in seconds. If the induced electric field outside the solenoid is 11 V/m a distance 2.0 m from the axis of the solenoid, find the radius of the solenoid.
Solution: Given
When there is a changing magnetic field through a coil, it causes an induced emf in the coil, which is based on the Faraday's law of electromagnetic induction. Further, the induced emf is related to the induced electric field along the circumference of the coil.
Magnetic field inside the solenoid, B =5.0×t T
Induced electric field at radial distance (from the axis of solenoid) r = 2.0 m is E =11 V/m
Radius (R) of solenoid =?
The induced electric field at the radial distance r from the axis of the solenoid is expressed as:
E×(2πr) =
E×(2πr) =
E×(2×r) =R2
E×(2×r) = R2
11×(2×2) =R2×5
R=2.966 
3.0 m
Q7) A coil has 1000 turns and 500 cm2 as it's area. The plane of the coil is placed at right angles to a magnetic induction field of 2×10−5Wb/m2. The coil is rotated through 1800 in 0.2 seconds. What is the average emf induced in the coil (in milli volts)?
Solution:
The induced emf will be given by
emf=
=
=
=
=10×10−3V=10 mV
Q8) In a uniform magnetic field of induction B, a wire in the form of semicircle of radius r rotates about the diameter of the circle with angular velocity ω. If the total resistance of the circuit is R, then find the mean power generated per period of rotation.
Figure
Solution:
Q9) Explain Faraday’s Experiment? What conclusion faraday draws from his experiment?
Solution: Faraday’s law of electromagnetic induction (referred to as Faraday’s law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). This phenomenon is known as electromagnetic induction.
Faraday’s law states that a current will be induced in a conductor which is exposed to a changing magnetic field. Lenz’s law of electromagnetic induction states that the direction of this induced current will be such that the magnetic field created by the induced current opposes the initial changing magnetic field which produced it. The direction of this current flow can be determined using Fleming’s right-hand rule.
Let us understand Faraday’s law by faraday‘s experiment.
FARADAY’S EXPERIMENT
In this experiment, Faraday takes a magnet and a coil and connects a galvanometer across the coil. At starting, the magnet is at rest, so there is no deflection in the galvanometer i.e. the needle of the galvanometer is at the center or zero position. When the magnet is moved towards the coil, the needle of the galvanometer deflects in one direction.
Figure 1
When the magnet is held stationary at that position, the needle of galvanometer returns to zero position. Now when the magnet moves away from the coil, there is some deflection in the needle but opposite direction, and again when the magnet becomes stationary, at that point respect to the coil, the needle of the galvanometer returns to the zero position.
Similarly, if the magnet is held stationary and the coil moves away, and towards the magnet, the galvanometer similarly shows deflection. It is also seen that the faster the change in the magnetic field, the greater will be the induced EMF or voltage in the coil.
Let us take a look at the following table. This table is showing the observation of faraday’s experiment.
Position of magnet | Deflection in galvanometer |
Magnet at rest | No deflection in the galvanometer |
Magnet moves towards the coil | Deflection in galvanometer in one direction |
Magnet is held stationary at same position (near the coil) | No deflection in the galvanometer |
Magnet moves away from the coil | Deflection in galvanometer but in the opposite direction |
Magnet is held stationary at the same position (away from the coil) | No deflection in the galvanometer |
CONCLUSION
From this experiment, Faraday concluded that whenever there is relative motion between a conductor and a magnetic field, the flux linkage with a coil changes and this change in flux induces a voltage across a coil.
Michael Faraday formulated two laws on the basis of the above experiments. These laws are called Faraday’s laws of electromagnetic induction.
Q10) What is Faraday’s Law? Derive its expression and also give factor affecting induced emf ?
Solution: Faraday’s First Law
Any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil. This emf induced is called induced emf and if the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current.
Method to change the magnetic field:
Faraday’s Second Law
It states that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil. The flux linkage of the coil is the product of the number of turns in the coil and flux associated with the coil.
Faraday Law Formula
Consider, a magnet is approaching towards a coil. Here we consider two instants at time T1 and time T2. Coil has N number of turns.
Flux linkage with the coil at time,
………..(1)
Flux linkage with the coil at time,
………..(2)
Change in flux linkage,
………..(3)
Let this change in flux linkage be,
………..(4)
So, the Change in flux linkage
………..(5)
Now the rate of change of flux linkage
………..(6)
Take derivative on right-hand side we will get as N is number of turns of coils which is constant so taken out of derivative, we get
………..(7)
The rate of change of flux linkage
………..(8)
Where E is induced emf.
But according to Faraday’s law of electromagnetic induction, the rate of change of flux linkage is equal to induced emf.
………..(9)
Considering Lenz’s Law
Where:
Factor Affecting Induced Emf
Q11) Explain Ampere’s circuital law?
Solution:
AMPERE’S LAW
Ampere’s circuital law (ACL) relates current to the magnetic field associated with the current.The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source.
Figure 1
Ampere's Law states that
The integral around a closed path of the component of the magnetic field tangent to the direction of the path equals μ0 times the current intercepted by the area within the path.
= μ0I ………..(1)
Ampere's Law tells that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.
The equation (1) can be written as
= I ………..(2)
Thus the line integral (circulation) of the magnetic field around some arbitrary closed curve is proportional to the total current enclosed by that curve.
InordertoapplyAmpère’sLaw
Allcurrentshavetobesteady(i.e.donotchangewithtime)
Onlycurrentscrossingtheareainsidethepatharetakenintoaccountandhavesomecontributiontothemagneticfield
Currentshavetobetakenwiththeiralgebraicsigns(thosegoing“out”ofthesurfacearepositive,thosegoing“in”arenegative)-userighthand’sruletodeterminedirectionsandsigns.
The total magnetic circulation is zero only in the following cases:-
Ampère’sLawcanbeusefulwhencalculatingmagneticfieldsofcurrentdistributionswithahighdegreeofsymmetry.
Finally, be aware that the form of Ampère’s Law addressed here applies to magnetostatics only. In the presence of a time-varying electric field, the right side of ACL includes an additional term known as the displacement current.
Ampere's Circuital Law
Consider a long thin wire carrying a steady current I. Suppose that the wire is orientated such that the current flows along the z-axis. Consider some closed loop C in the x-y plane which circles the wire in an anti-clockwise direction, looking down the z-axis. Suppose that dr is a short straight-line element of this loop. Let us form the dot product of this element with the local magnetic field 
. Thus,
……....(3)
Where θ is the angle subtended between the direction of the line element and the direction of the local magnetic field. We can calculate a dw for every line element which makes up the loop C. If we sum all of the dw values thus obtained, and take the limit as the number of elements goes to infinity, we obtain the line integral
…….....(4)
What is the value of this integral? In general, this is a difficult question to answer. However, let us consider a special case. Suppose that 
is a circle of radius 
centred on the wire. In this case, the magnetic field-strength is the same at all points on the loop. In fact,
……......(5)
Moreover, the field is everywhere parallel to the line elements which make up the loop. Thus,
…….......(6)
or
……….....(7)
In other words, the line integral of the magnetic field around some circular loop C, centred on a current carrying wire, and in the plane perpendicular to the wire, is equal to μ0 times the current flowing in the wire. Note that this answer is independent of the radius r of the loop: i.e., the same result is obtained by taking the line integral around any circular loop centred on the wire.
In 1826, Ampère demonstrated that Eq. (7) holds for any closed loop which circles around any distribution of currents.
Determining Magnetic Field by Ampere’s Law
Suppose you have a long enough wire that carries a constant current I in amps. How would you determine the magnetic field wrapping the wire at any distance r from the wire?
In the figure below ( Figure 2), a long wire exists that carries current in Amps. We need to find out how much is the magnetic field at a distance r. Therefore, we sketch an imaginary route around the wire indicated by dotted blue toward the right in the figure.
Figure 2
According to the second equation, if the magnetic field is integrated along the blue path, then it has to be equal to the current enclosed, I.
The magnetic field doesn’t vary at a distance r due to symmetry. The path length (in blue) in figure 2 equals to the circumference of a circle, 2πr.
When a constant value H is added to the magnetic field, the equation’s left side looks like this:
We have figured the magnitude of the field H. Since r is arbitrary, the value of the field H is known.
According to the equation, the magnetic field lowers in magnitude as we move wider. Hence, Ampere’s law can be applied to calculate the extent of the magnetic field surrounding the wire. The field H is a Vector field which reveals that each region has both a direction and a magnitude. The field’s direction is tangential at every point to the imaginary loops as shown in figure 3 and the right-hand rule finds the direction of the magnetic field.
Figure 3
Q12) Discuss concept of displacement current?
Or
What correction is done by Maxwell to Ampere’s law?
Solution: Displacement current is the rate of change of electric displacement field.
The type of current which passes through a conductor is known as conduction current and is caused by the actual movement of electrons through the conductor. This type of current is mostly used in our day to day life. There is also another kind of current, which is known as displacement current. Displacement current differs from the conduction current because the displacement current does not involve electrons' movement. The displacement current has enormous importance for the propagation of electromagnetic waves. In electromagnetism, displacement current is a quantity appearing in Maxwell's equations that is defined in terms of the rate of change of electric displacement field.
Displacement current has the units of electric current density, and it has an associated magnetic field just as actual currents do.
It is mathematically represented as
ID = JD S = S 
Where
S = Area of the plate of the capacitor
ID =Displacement current
JD =Displacement current density
D = εE where ε is permittivity of the medium
During the charging and discharging process of the capacitor, the electric current flows through the wires of the circuit. However, no current flow between the plates of the capacitor.
According to Ampere's law, the magnetic field should not present between the plates as there is no current, but in reality, the magnetic field exists there. Maxwell formulated this limitation of Ampere's law by adding a term in the equation of Ampere's law to solve the issue.
Maxwell predicted that the magnetic field will still exist even in the absence of conduction current, and the magnetic field may be associated with the changing electric field. This theory of Maxwell was experimentally proved.
Since the magnetic field is associated with the electric field, the general displacement current formula is given by,
= μ0( I+ε0
)
This equation is the generalized formula of Maxwell-Ampere law.
Displacement Current Definition
The displacement current (ID) is the part which Maxwell has added to the Ampere's law.
ID = ε0
ε0= Permittivity of free space
Derivative of electric flux
Electric flux is the time rate change of flow of the electric field through a surface. If we take the derivative of electric flux, we get the rate of change of the electric field of a given area concerning time.
Q13) If the electric field strength of a radio broadcast signal at a TV receiver is given by
E = 5.0 cos (t-y) az, V/m,
Determine the displacement current density. If the same field exists in a medium whose conductivity is given by 2.0 x 103(mho)/cm, find the conduction current density.
Solution: E at a TV receiver in free space= 5.0 cos (t-y) az, V/m
Electric flux density
D =0E = 50cos (t-y) az, V/m
The displacement current density
Jd= 
= 
Jd=-50sin (t-y) az, V/m2
The conduction current density,
Jc =E
=2.0 x 103(mho) /cm
= 2 x 105mho /m
Jc= 2 x 105x 5 cos (t-y) az
Jc= 106cos (t-y) az V/m2
Q14) Given E = 10 sin (t-y) ay V/m, in free space, determine D, B and H.
Solution: E = 10 sin (t-y) ay, V/m
D =0E,
0= 8.854 x 10-12 F/m
D = 100sin (t-y) ay, C/m2
Second Maxwell’s equation is
x E = -B
That is,
or
As Ey= 10 sin (t-z) V/m
= 0
Now,x E becomes
x E = - 
ax
= 10cos (t-z) ax
= - 
Q15) If the electric field strength, E of an electromagnetic wave in free space is given by
E = 2 cos (t - 
) ay V/m,
Find the magnetic field, H
Solution: We have
B/t =-x E
Thus
Also
Q16) What do you meant by gradient? Discuss its physical significance?
Solution: The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that
The term "gradient" is typically used for functions with several inputs and a single output (a scalar field). Yes, you can say a line has a gradient (its slope), but using "gradient" for single-variable functions is unnecessarily confusing. Keep it simple.
“Gradient” can refer to gradual changes of colour, but we’ll stick to the math definition. You’ll see the meanings are related.
PHYSICAL SIGNIFICANCE OF GRADIENT
A scalar field may be represented by a series of level surfaces each having a stable value of scalar point function θ. The θ changes by a stable value as we move from one surface to another. These surfaces are known as Gaussian surfaces. Now let the two such surfaces are very close together, be represented by two scalar point functions and (θ + d θ). Let ‘r’ and (r + d θ) be the position vectors of points A and B, on the surfaces θ and (θ + d θ) correspondingly with respect to an origin 0 as shown in Figure. Clearly, the vector AB will be dr. Let the least detachment between the two surfaces ‘dn’ be in the direction of unit usual vector n at A.
9 Figure
dn = drcos θ
= | n | dr | cosθ =n .dr
Dϕ = ∂ϕ/ dn =∂ϕ/dn n .dr…… (1)
Since the continuous scalar function defining the level surfaces (Gaussian surfaces) has a value θ at point A (x, y, z) and (θ + dθ) at point (x + dx, y + dy, z + dz), we have
dϕ = ∂ ϕ/dx dx +dϕ/∂y + ∂ϕ/∂x dz
= (I ∂ϕ/∂x +j ∂ϕ/vy +k ∂ϕ/∂z) .(idx +jdy +kdz)
= ∆ ϕ. dr … … (2)
From equations (1) and (2), equating the values of d θ,
We obtain ∆θ .dr =∆ϕ= ∂ϕ/ ∂n n .dr
As dr is an arbitrary vector, we have
∆ϕ =∂ϕ/∂ n
Grad ϕ = ∂ ϕ/∂n n
Therefore, the gradient an of a scalar field at any point is a vector field, the scale of which is equal to the highest rate of increase of θ at that point and the direction of it is similar as that of usual to the level surface at that point.
Its physical significance of grad can be understand in terms of the graph of some function z = f(x, y), where f is a reasonable function – say with continuous first partial derivatives. In this case we can think of the graph as a surface whose points have variable heights over the xy – plane. An illustration is given below.
10 Figure
If, say, we place a marble at some point (x, y) on this graph with zero initial force, its motion will trace out a path on the surface, and in fact it will choose the direction of steepest descent. This direction of steepest descent is given by the negative of the gradient of f. One takes the negative direction because the height is decreasing rather than increasing.
Using the language of vector fields, we may restate this as follows: For the given function f(x, y), gravitational force defines a vector field F over the corresponding surface z = f(x, y), and the initial velocity of an object at a point (x, y) is given mathematically by –∇f(x, y). The gradient also describes directions of maximum change in other contexts. For example, if we think offas describing the temperature at a point(x, y), then the gradient gives the direction in which the temperature is increasing most rapidly.
Q17) Define Gauss’s law? What is the value of Gauss’s law in free space?
Solution: The law was published posthumously in 1867 as part of a collection of work by the famous German mathematician Carl Friedrich Gauss.
Gauss’s law for the electric field describes the static electric field generated by a distribution of electric charges.
It states that the electric flux through any closed surface is proportional to the total electric charge enclosed by this surface.
let’s calculate the electric flux through a spherical surface around a positive point charge q, since we already know the electric field in such a situation. Recall that when we place the point charge at the origin of a coordinate system, the electric field at a point P that is at a distance r
from the charge at the origin is given by
is the radial vector from the charge at the origin to the point P. We can use this electric field to find the flux through the spherical surface of radius r, as shown in Figure 2
.
Figure 1: A closed spherical surface surrounding a point charge q.
Then we apply Φ=∫S
⋅
dA
to this system and substitute known values. On the sphere, 
and r=R
so for an infinitesimal area dA,
We now find the net flux by integrating this flux over the surface of the sphere
where the total surface area of the spherical surface is 4πR2. This gives the flux through the closed spherical surface at radius ras
Φ=q/ϵ0
A remarkable fact about this equation is that the flux is independent of the size of the spherical surface. This can be directly attributed to the fact that the electric field of a point charge decreases as 1/r2with distance, which just cancels the r2 rate of increase of the surface area.
Gauss' Law is the first of Maxwell's Equations which dictates how the Electric Field behaves around electric charges. Gauss' Law can be written in terms of the Electric Flux Density and the Electric Charge Density as:
∇·E = ρ/ε0
0r
∇·D = ρ
Hence, Gauss' law is a mathematical statement that the total Electric Flux exiting any volume is equal to the total charge inside. Hence, if the volume in question has no charge within it, the net flow of Electric Flux out of that region is zero. If there is positive charge within a volume, then there exists a positive amount of Electric Flux exiting any volume that surrounds the charge. If there is negative charge within a volume, then there exists a negative amount of Electric Flux exiting (i.e. the Electric Flux enters the volume).
In integral form
in differential form
∇·E = ρ/ε0
But in free space charge ρ is zero so ∇·E =0
Q18) Derive electromagnetic wave equation?
Solution: Wave is nothing but a pattern of disturbance which propagates and carry energy with it. You can produce a wave on a rope by moving one end of the rope up and down. The wave produces on rope needs a medium to propagate and here medium is rope itself. This type of waves is known as mechanical waves. But in the case of Electromagnetic waves, they don't need a medium to propagate. Electromagnetic waves are waves that are created as a result of variations of electric field and a magnetic field. Or we can say that Electromagnetic waves are nothing but changing magnetic and electric fields. Electromagnetic waves are also known to be solutions of Maxwell's equations. And Maxwell's equations are the fundamental equations of electrodynamics. Electromagnetic waves can transmit energy and travel through a vacuum.
Light waves are examples of electromagnetic waves. Generally, Electromagnetic waves are shown by a sinusoidal graph.
Figure 1
As shown in figure Electromagnetic waves consist of time-varying electric and magnetic fields and they are perpendicular to each other and these both fields are also perpendicular to the direction of propagation of waves.
Maxwell’s four equations are given by
∇·E = ρ/ε0 (1)
∇×E = −∂B/∂t (2)
∇×H = J + ∂D/∂t (3)
∇·B = 0 (4)
These equations illustrate the unique coexistence in nature of the electric field and the magnetic field. The first two equations give the value of the given flux through a closed surface, and the second two equations give the value of a line integral around a loop. In this notation,
∇=(∂/∂x, ∂/∂y, ∂/∂z)
E is the electric vector
B is the magnetic induction
ρ is the electric charge density
j is the electric current density
ε0 is the permittivity of free space
c is the speed of light.
In addition to Maxwell equations, the following identities are useful:
J = σE (5)
D = εE (6)
B = μH (7)
Here,
D is the electric displacement
H is the magnetic vector
σ is the specific conductivity
ε is the dielectric constant (or permittivity)
μ is the magnetic permeability
Free space or non-conducting medium. We know that non conducting medium means no current so conductivity is zero i.e. σ=0
So current density J=σE will also become zero as σ=0 Also free space means no charges which leads to ρ=0. These points mentioned below.
(a) No condition current i.eσ=0, thus J=0 ( J=σE)
(b) No charges (i.eρ=0)
For the case of no charges or currents, that is, j = 0 and ρ = 0, and a homogeneous medium. Using these the Maxwell equation can rewritten as
∇.D=0 or ∇.E=0 as ρ=0 (12)
∇ x E= -dB/dt or ∇ x E= -μdH/dt because B = μH (13)
∇x H=d D/dt or ∇ x H = εdE/dt (J=0) and D = εE (14)
∇.B=0 (15)
WAVE EQUATION IN TERMS OF ELECTRIC FIELD INTENSITY, E
Now taking curl of second Maxwell’s equation (13) ,we get
∇x(∇ x E)=- μ d/dt (∇ x H)
Applying standard vector identity, that is [∇ *(∇*E)=∇(∇.E)-∇2E] on left hand side of above equation, we get
∇ (∇ .E)-∇2E= -μ d/dt (∇x H) (16)
Substituting equations (13) and (14) in equations (16) ,we get
-∇2E= – μεd/dt (dE/dt)
Or ∇2E=με d 2 E/dT2 (17)
Equation (17) is the required wave equation in terms of electric field intensity, E for free space . This is the law that E must obey.
WAVE EQUATION IN TERMS OF MAGNETIC FIELD INTENSITY, H
Take curl of fourth Maxwell’s equation (14) ,we get
∇x(∇xH)=ε d/dt(∇xE)
Applying standard vector identity that is
[∇*(∇*H)=∇ (∇.H)-∇2H]
On left side of above equation , we get
∇(∇.H)-∇2H= ε d/dt(∇xE) (18)
Substituting equations (14) and (13) in equation(18) ,we get
-∇2H= – μεd/dt(dH/dt)
Or
∇2H=με d2H/dt2 (19)
Equations (19) is the required wave equation in terms of magnetic field intensity, H and this is the law that H must obey
For vacuum μ=μ0 and ε=ε0, equations (17) and (19) will become
∇2 E=μ0ε0 d2E/dt2 (20)
And ∇2H= μ0ε0 d2H/dt2 (21)
This leads to an expression for the velocity of propagation
From equation both equations (20) and (21) have the form of the general wave equation for a wave
(x,t) traveling in the x direction with speed v. Equating the speed with the coefficients, we derive the speed of electric and magnetic waves, which is a constant that we symbolize with “c”
It is useful to note that in vacuum
c2=1/ε0μ0
Where μ0 is the permeability of free space
Let us rewrite the equation (20) and (21) for one dimension.
Or d 2 E/dx2 = μ0ε0 d 2 E/dt2 (22)
d2H/dx2= μ0ε0 d2H/dt2 (23)
The simplest solutions to the differential equations (22) and (23) are sinusoidal wave functions:
E(x) = Emaxcos(kx-t) (24)
B(x) = Bmaxcos(kx-t) (25)
where k = 2π/λ is the wavenumber , ω = 2πƒ is the angular frequency, λ is the wavelength, f is the frequency and ω /k=f=v= c.
Q19) State Gauss Divergence Theorem ?
Solution: GAUSS DIVERGENCE THEOREM
In vector calculus, divergence theorem is also known as Gauss’s theorem. It relates the flux of a vector field through the closed surface to the divergence of the field in the volume enclosed.
The Gauss divergence theorem states that the vector’s outward flux through a closed surface is equal to the volume integral of the divergence over the area within the surface.
The sum of all sources subtracted by the sum of every sink will result in the net flow of an area. Gauss divergence theorem is the result that describes the flow of a vector field by a surface to the behaviour of the vector field within it.
Let 
be a smooth vectorfield defined on a solid region V with boundary surface A oriented outward. Gauss divergence theorem is given by
The divergence of a vector field F, denoteddiv(F) or
(the notation used in this work), is defined by a limit of the surface integral
| (1) |
where the surface integral gives the value of F integrated over a closed infinitesimal boundary surface S=
surrounding a volume element V, which is taken to size zero using a limiting process. The divergence of a vector field is therefore a scalar field. If 
=0, then the field is said to be a divergenceless field. The symbol
is variously known as "nabla" or "del."
The physical significance of the divergence of a vector field is the rate at which "density" exits a given region of space. The definition of the divergence therefore follows naturally by noting that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region. By measuring the net flux of content passing through a surface surrounding the region of space, it is therefore immediately possible to say how the density of the interior has changed. This property is fundamental in physics, where it goes by the name "principle of continuity." When stated as a formal theorem, it is called the divergence theorem, also known as Gauss's theorem.
Q20) State Stokes Theorem?
Solution: Stokes Theorem
Stokes Theorem is also referred to as the generalized Stokes Theorem. It is a declaration about the integration of differential forms on different manifolds. It generalizes and simplifies the several theorems from vector calculus. According to this theorem, a line integral is related to the surface integral of vector fields.
Figure 7
Stokes’ theorem relates the surface integral of the curl of the vector field to a line integral of the vector field around some boundary of a surface. It is named after George Gabriel Stokes. Although the first known statement of the theorem is by William Thomson and it appears in a letter of his to Stokes.
Stoke’s theorem statement is “the surface integral of the curl of a function over the surface bounded by a closed surface will be equal to the line integral of the particular vector function around it.” Stokes theorem gives a relation between line integrals and surface integrals. Depending upon the convenience, one integral can be computed in terms of the other.
Stokes Theorem Formula:
It is,
∮C 
= ∬S (∇ ×
). 
Where,
C = A closed curve.
S = Any surface bounded by C.
F = A vector field whose components are continuous derivatives in S.
Thus, it means that if we walk in the positive direction around C with our head pointing in the direction of n, then the surface will always be on our left. S is oriented smooth surface bounded by a simple, closed smooth-boundary curve C with positive orientation. It can be noted that the surface S can actually be any surface so long as its boundary curve is given by C. This is something that can be used to our advantage to simplify the surface integral on the occasion.
Stokes theorem physics and stokes theorem mathematics are very popular. We can find many applications of stokes theorem in Physics. It helps to derive many useful formulae and equations. For example, stokes theorem in electromagnetic theory is very popular in Physics.
Q21) What is Maxwell’s Equation in Differential and Integral Form?
Solution: Maxwell Equations
Maxwell equations are of fundamental importance since they describe the whole of classical electromagnetic phenomena.
Maxwell's Equations are a set of 4 complicated equations that describe the world of electromagnetics. These equations describe how electric and magnetic fields propagate, interact, and how they are influenced by objects.
Maxwell was one of the first to determine the speed of propagation of electromagnetic (EM) waves was the same as the speed of light - and hence to conclude that EM waves and visible light were really the same thing.
These equations are rules the universe uses to govern the behaviour of electric and magnetic fields. A flow of electric current will produce a magnetic field. If the current flow varies with time (as in any wave or periodic signal), the magnetic field will also give rise to an electric field. Maxwell's Equations shows that separated charge (positive and negative) gives rise to an electric field - and if this is varying in time as well will give rise to a propagating electric field, further giving rise to a propagating magnetic field.
From a classical perspective, light can be described as waves of electromagnetic radiation. As such, Maxwell equations are very useful to illustrate a number of the characteristics of light including polarization.
Maxwell’s first equation (Gauss' Law for Electric Fields)
Gauss' Law is the first of Maxwell's Equations which dictates how the Electric Field behaves around electric charges. Gauss' Law can be written in terms of the Electric Flux Density and the Electric Charge Density as:
∇·E = ρ/ε0
0r
∇·D = ρ
Hence, Gauss' law is a mathematical statement that the total Electric Flux exiting any volume is equal to the total charge inside. Hence, if the volume in question has no charge within it, the net flow of Electric Flux out of that region is zero. If there is positive charge within a volume, then there exists a positive amount of Electric Flux exiting any volume that surrounds the charge. If there is negative charge within a volume, then there exists a negative amount of Electric Flux exiting (i.e. the Electric Flux enters the volume).
The First Maxwell’s equation (Gauss’s law for electricity)
The Gauss’s law states that flux passing through any closed surface is equal to 1/ε0 times the total charge enclosed by that surface.
Integral form of Maxwell’s 1st equation
It is the integral form of Maxwell’s 1st equation.
Maxwell’s first equation in differential form
It is called the differential form of Maxwell’s 1st equation.
The Second Maxwell’s equation (Gauss’s law for magnetism)
The Gauss’s law for magnetism states that net flux of the magnetic field through a closed surface is zero because monopoles of a magnet do not exist.
we know that Gauss' Law for Electric Fields states that the divergence of the Electric Flux Density D is equal to the volume electric charge density. But the second equation, Gauss' Magnetism law states that the divergence of the Magnetic Flux Density (B) is zero.
Why? Why isn't the divergence of B equal to the magnetic charge density?
Well - it is. But it just so happens that no one has ever found magnetic charge - not in a laboratory or on the street or on the subway. And therefore, until this hypothetical magnetic charge is found, we set the right side of Gauss' Law for Magnetic Fields to zero:
∇·B = 0
Now, you may have played with magnets when you were little, and these magnetic objects attracted other magnets similar to how electric charges repel or attract like electric charges. However, there is something special about these magnets - they always have a positive and negative end. This means every magnetic object is a magnetic dipole, with a north and South Pole. No matter how many times you break the magnetic in half, it will just form more magnetic dipoles. Gauss' Law for Magnetism states that magnetic monopoles do not exist - or at least we haven't found them yet.
Maxwell’s Third equation (Faraday's Law)
Faraday figured out that a changing Magnetic Flux within a circuit (or closed loop of wire) produced an induced EMF, or voltage within the circuit. Maxwell gives his second equation from this.
Faraday's Law tells us that a magnetic field that is changing in time will give rise to a circulating E-field. This means we have two ways of generating E-fields - from Electric Charges (or flowing electric charge, current) or from a magnetic field that is changing.
∇×E = −∂B/∂t
According to Faraday’s law of electromagnetic induction
It is the differential form of Maxwell’s third equation.
Maxwell’s fourth equation (Ampere's Law)
Ampere's Law tells us that a flowing electric current gives rise to a magnetic field that circles the wire. In addition to this, it also says that an Electric Field that is changing in time gives rise to a magnetic field that encircles the E-field - this is the Displacement Current term that Maxwell himself introduced.
This means there are 2 ways to generate a solenoidal (circulating) H-field - a flowing electric current or a changing Electric Field. Both give rise to the same phenomenon.
The modified form of ampere Law is given by
H.dL=(J+dD/dt)
Take integration on both sides we have
∫H.dL=∫(J+dD/dt).dS
Apply stoke’s theorem to L.H.S. of above equation, we get
∫( ∇xH).dS=∫ H.dL
Now the above equation is written as
∫( ∇xH).dS =∫(J+dD/dt).dS
By cancelling the surface integral on both sides we have
∇xH =J+dD/dt
Which is the differential form of Maxwell’s equation.
This can also be written in the form B
The Ampere-Maxwell Law
Begin with the Ampere-Maxwell law in integral form.
Set the equation to 0.
Q22) Give Physical Significance of Maxwell’s equation ?
Solution: Physical significance of Maxwell’s Ist equation
∇·E = ρ/ε0
According to this total electric flux through any closed surface is 1/0 times the total charge enclosed by the closed surfaces, representing Gauss's law of electrostatics, As this does not depend on time, it is a steady state equation. Here for positive , divergence of electric field is positive and for negative divergence is negative. It indicates that is scalar quantity.
Physical significance of Maxwell’s 2nd equation
∇·B = 0
It represents Gauss law of magnetostatic as ∇·B = 0 resulting that isolated magnetic poles or magnetic monopoles cannot exist as they appear only in pairs and there is no source or sink for magnetic lines of forces. It is also independent of time i.e. steady state equation.
Physical significance of Maxwell’s 3rd equation
∇×E = −∂B/∂t
It shows that with time varying magnetic flux, electric field is produced in accordance with Faraday is law of electromagnetic induction. This is a time dependent equation.
Physical significance ofMaxwell’s 4th equation
∇×H = J + ∂D/∂t
This is a time dependent equation which represents the modified differential form of Ampere's circuital law according to which magnetic field is produced due to combined effect of conduction current densityand displacement current density.
