Test: Maxwell's Equations Interpretation - Electronics and Communication Engineering (ECE) MCQ

• When a conductor moves in a magnetic field, an electromotive force (EMF) is induced in the conductor according to Faraday's law of electromagnetic induction. The direction of the induced current depends on the relative directions of the magnetic field and the motion of the conductor.
• According to Lenz's law, the induced current flows in a direction that opposes the change in magnetic flux through the conductor. This means that the induced current creates a magnetic field that opposes the external magnetic field causing the induction. The direction of the induced current can be determined using the right-hand rule.

Where N = number of turns, dΦ = change in magnetic flux, and e = induced e.m.f.
The negative sign says that it opposes the change in magnetic flux which is explained by Lenz law.

• If the magnetic field and the motion of the conductor are parallel (or antiparallel), no current is induced.
• If the magnetic field and the motion of the conductor are perpendicular, the induced current flows in a direction perpendicular to both the magnetic field and the motion of the conductor, according to the right-hand rule.
• So, the direction of the magnetic field and the direction of motion of the conductor are the primary factors determining the direction of the induced current.
• The other options listed (number of turns and length of the conductor, strength of the magnetic field, and speed of movement of the conductor) may affect the magnitude of the induced current but do not determine its direction.

The energy density in a magnetic field is given by the formula:
u = BH/2
where:

• u is the energy density in joules per cubic meter
• B is the flux density in teslas
• H is the magnetic field strength in amperes per meter

Therefore, the correct answer is option 4, BH/2.
Here is a brief explanation of why the other options are incorrect:

• Option 1, BH2/2, is the energy density in the magnetic field of a free space.
• Option 2, BH, is the force per unit length on a conductor carrying a current in a magnetic field.
• Option 3, BH2, is the energy density in the magnetic field of a material with a relative permeability of 1

Given
P = Number of poles = 6
N = Total turns = 6 × 500 = 3000 (since 500 turns per pole in series)
Total initial flux = 6 × 0.03 = 0.18 Wb. (since number of poles is 6)
Total residual flux = 6 × 0.003 = 0.018 Wb
Change in flux, dϕ = 0.18 - 0.018 = 0.162 Wb
Time of opening of circuit, dt = 0.02 second
From concept,
Induced EMF (E) = 
= 24300 V
Its direction is the same as the initial direction of the exciting current.

Time-varying fields are fields that change over time. Examples include temperature, pressure, electrical current, and magnetic fields. They are used in a variety of disciplines, such as engineering, physics, and meteorology. They can be used to measure and analyze the behavior of physical systems and can help predict future events.
A time-varying magnetic field causes flux to fluctuate, resulting in an electric field.
Maxwell’s equations integral form explain how the electric charges and electric currents produce magnetic and electric fields. The equations describe how the electric field can create a magnetic field and vice versa.
In time-varying fields:
electric scalar potential V and magnetic vector potential A satisfy the wave equation.
given as:

Hence option 1 is correct

Dynamically induced EMF: When the conductor is rotating and the field is stationary, then the emf induced in the conductor is called dynamically induced EMF.
Ex: DC Generator, AC generator
Static induced EMF: When the conductor is stationary and the field is changing (varying) then the emf induced in the conductor is called static induced EMF.
Ex: Transformer

• The sinusoidal components of a complex wave are called harmonic
• The lowest possible frequency of standing wave pattern is known as the fundamental frequency or the first harmonic.
• The harmonic with the lowest frequency is called fundamental harmonic.
• Frequency of the complex wave is the same as that of the first harmonic of that wave.
•  All non-sinusoidal waves can be broken up into a series of sinusoidal waves whose frequencies are integral multiples of the frequency of the fundamental wave. 
• Each harmonic is a pure sinusoid. Waves having 2f, 4f, 6f etc. are called even harmonics and those having frequencies 3f 5f, 7f, etc. are called odd harmonics.
• A signal with half-wave symmetry doesn’t have a DC offset or average value  nor any even harmonic only odd harmonics. In general, such a signal will contain all the odd harmonics.

Faraday's first law of electromagnetic induction:
It states that whenever a conductor is placed in a varying magnetic field, emf is induced which is called induced emf. If the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current.
​Faraday's second law of electromagnetic induction:
It states that the magnitude of the voltage 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 number of turns in the coil and flux associated with the coil.​

Where N = number of turns, dΦ = change in magnetic flux and v = induced voltage.
The negative sign says that it opposes the change in magnetic flux which is explained by Lenz law.

The correct Maxwell's equation is: 
Maxwell's Equations for time-varying fields is as shown:

• Laplace's law indicates that the tension on the wall of a sphere is the product of the pressure times the radius of the chamber and the tension is inversely related to the thickness of the wall. Therefore the option 1 is incorrect.

• According to Lenz's law, the direction of induced emf or current in a circuit is such as to oppose the cause that produces it. Therefore the option 2 is correct.
• Fleming's right-hand rule shows the direction of induced current but it gives no relation between the direction of induced emf or current in a circuit is such as to oppose the cause that produces it. Therefore the option 3 is incorrect.
• This law is also known as loop rule or voltage law (KVL) and according to it “the algebraic sum of the changes in potential in a complete traversal of a mesh (closed-loop) is zero”, i.e. Σ V = 0. Therefore the option 3 is incorrect.

Faraday's laws: Faraday performed many experiments and gave some laws about electromagnetism.
Faraday's First Law:
Whenever a conductor is placed in a varying magnetic field an EMF gets induced across the conductor (called induced emf), and if the conductor is a closed circuit then induced current flows through it.
A magnetic field can be varied by various methods:

• By moving magnet
• By moving the coil
• By rotating the coil relative to a magnetic field

Faraday's second law of electromagnetic induction states that the magnitude of induced emf is equal to the rate of change of flux linkages with the coil.
According to Faraday's law of electromagnetic induction, the rate of change of flux linkages is equal to the induced emf: