Maxwell's Equations MCQ Quiz - Objective Question with Answer for Maxwell's Equations - Download Free PDF

Faraday's Laws of Electromagnetic Induction

Definition: Faraday's Laws of Electromagnetic Induction describe how an electromotive force (EMF) is induced in a conductor when the magnetic flux linked with it changes. This principle is fundamental to the operation of many electrical devices, including electric generators, transformers, and induction motors.

Working Principle: According to Faraday's first law, an EMF is induced in a conductor when there is a change in the magnetic flux linked with it. Faraday's second law quantifies this phenomenon, stating that the magnitude of the induced EMF is proportional to the rate of change of magnetic flux through the conductor.

Mathematical Representation:

The induced EMF (ε) can be expressed as:

ε = -dΦ/dt

Where:

• ε: Induced EMF (volts)
• Φ: Magnetic flux (Weber)
• t: Time (seconds)

The negative sign in the equation represents Lenz's Law, which states that the induced EMF always opposes the change in magnetic flux that causes it.

Correct Option Analysis:

The correct option is:

Option 4: When there is relative motion between a conductor and a magnetic field.

When a conductor moves relative to a magnetic field, or when the magnetic field changes around a stationary conductor, the magnetic flux linked with the conductor changes. According to Faraday's Laws, this change in magnetic flux induces an EMF in the conductor. This is the fundamental principle behind the operation of electric generators, where the relative motion between a coil of wire (conductor) and a magnetic field generates electricity.

Example:

Consider a simple setup where a conductor is moved through a magnetic field:

• When the conductor moves through the magnetic field, the magnetic flux linking the conductor changes.
• This changing flux induces an EMF in the conductor, which can drive a current if the circuit is closed.

This principle is utilized in generators, where mechanical energy is converted into electrical energy by rotating a coil within a magnetic field, causing a relative motion and inducing an EMF.

Important Information:

To further understand the analysis, let’s evaluate the other options:

Option 1: When the temperature of the conductor increases.

An increase in temperature of a conductor does not induce an EMF. While temperature changes can affect the resistance of a conductor, they do not directly cause a change in magnetic flux, which is a prerequisite for the induction of EMF according to Faraday's Laws. Thus, this option is incorrect.

Option 2: When the conductor is stationary in a magnetic field.

If a conductor is stationary in a constant magnetic field, the magnetic flux linked with the conductor remains unchanged. As there is no change in magnetic flux, no EMF is induced. This principle highlights the necessity of relative motion or a changing magnetic field for the induction of EMF. Therefore, this option is also incorrect.

Option 3: When the magnetic field is constant.

A constant magnetic field does not induce an EMF in a stationary conductor. According to Faraday's Laws, an EMF is induced only when there is a change in magnetic flux. A constant magnetic field does not produce any such change, and hence, no EMF is induced. This makes this option incorrect as well.

Conclusion:

The induction of EMF in a conductor, as described by Faraday's Laws, requires a change in magnetic flux. This change can occur due to relative motion between the conductor and the magnetic field or due to a time-varying magnetic field. The correct option, therefore, is Option 4: "When there is relative motion between a conductor and a magnetic field." This principle is fundamental to the operation of many electrical machines and devices, making it a cornerstone of electromagnetic theory and practical applications in electrical engineering.

Explanation:

Induced EMF Calculation in a Coil:

Electromotive force (EMF) is generated in a coil when there is a change in magnetic flux through the coil. This phenomenon is explained by Faraday's Law of Electromagnetic Induction, which states that the induced EMF is proportional to the rate of change of magnetic flux.

Faraday's Law:

Mathematically, Faraday's Law can be expressed as:

EMF = -N × (ΔΦ / Δt)

Where:

• N: Number of turns in the coil
• ΔΦ: Change in magnetic flux (in Weber)
• Δt: Time interval during which the change occurs (in seconds)
• EMF: Electromotive force (in Volts)

The negative sign in the formula indicates Lenz's Law, which states that the induced EMF opposes the change in magnetic flux that produces it.

Given Data:

• Number of turns in the coil, N = 30
• Change in magnetic flux, ΔΦ = 0.1 Weber
• Time interval, Δt = 1 second

Step-by-Step Solution:

Using the formula for Faraday's Law:

EMF = -N × (ΔΦ / Δt)

Substitute the values:

EMF = -30 × (0.1 / 1)

Simplify the expression:

EMF = -30 × 0.1

EMF = -3 Volts

Explanation:

Maxwell's First Equation

Definition: Maxwell's first equation, also known as Gauss's law for electricity, describes the relationship between electric charge and electric flux density. It states that the divergence of the electric flux density (\( \overline{D} \)) is equal to the volume charge density (\( \rho_{v} \)). This fundamental equation is derived from the principles of electrostatics and is expressed mathematically as:

\(\rm \operatorname{div} \overline{D} =\rho_{v}\)

This equation is essential in understanding how electric charges create electric fields and how they interact with their surroundings.

Understanding the Terms:

• \(\rm \overline{D}\) (Electric Flux Density): Represents the amount of electric flux passing through a unit area. It is related to the electric field (\(\rm \overline{E}\)) and the permittivity of the medium (\(\varepsilon\)) as \(\rm \overline{D} = \varepsilon \overline{E}\).
• \(\rho_{v}\) (Volume Charge Density): Represents the amount of electric charge per unit volume.
• \(\rm \operatorname{div}\): The divergence operator calculates the net flux leaving a point in space.

Explanation of Maxwell's First Equation:

Maxwell's first equation states that the divergence of the electric flux density (\(\rm \operatorname{div} \overline{D}\)) at a point is equal to the volume charge density (\(\rho_{v}\)) at that point. Physically, it implies that electric charges act as sources or sinks of electric flux. Positive charges emit flux, while negative charges absorb flux.

In integral form, Gauss's law can be expressed as:

\(\int_{S} \overline{D} \cdot \overline{n} \, dS = Q_{enc}\)

Where:

• \(S\): A closed surface.
• \(\overline{n}\): Unit normal vector to the surface.
• \(Q_{enc}\): The total charge enclosed within the surface \(S\).

This equation highlights that the total electric flux through a closed surface is proportional to the charge enclosed within that surface.

Correct Option Analysis:

The correct option is:

Option 4: \(\rm \operatorname{div} \overline{D} =\rho_{v}\)

This option correctly represents Maxwell's first equation in its differential form. It accurately describes the relationship between the divergence of the electric flux density (\(\rm \operatorname{div} \overline{D}\)) and the volume charge density (\(\rho_{v}\)). This is a foundational expression in electromagnetism and forms the basis for understanding electric fields and their interaction with charges.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm \operatorname{div} D =\overline{\rho_{v}}\)

This option uses the term \(D\) without the vector notation (\(\overline{D}\)). In electromagnetism, the electric flux density is a vector quantity, and its divergence is calculated with respect to its components. Therefore, the omission of the vector notation makes this representation incomplete and incorrect.

Option 2: \(\rm \operatorname{div} \overline{D} =\overline{\rho_{v}}\)

This option incorrectly represents the volume charge density (\(\rho_{v}\)) as a vector quantity (\(\overline{\rho_{v}}\)). However, \(\rho_{v}\) is a scalar quantity representing the amount of charge per unit volume. Since divergence results in a scalar value, this representation is incorrect.

Option 3: \(\rm div D = \rho_{v}\)

This option omits the vector notation for the electric flux density (\(D\)). While the equation may appear correct in form, the lack of vector notation for \(D\) makes it incomplete, as \(D\) is a vector field. Proper representation requires the use of \(\rm \overline{D}\) to signify the vector nature of the electric flux density.

Conclusion:

Maxwell's first equation is a cornerstone of electromagnetism, describing the interaction between electric charges and the electric fields they produce. The correct representation of this equation in differential form is \(\rm \operatorname{div} \overline{D} =\rho_{v}\), as stated in option 4. This equation highlights the fundamental relationship between the divergence of the electric flux density and the volume charge density, providing essential insights into electric field behavior. Understanding the correct formulation of Maxwell's equations is critical for studying and applying electromagnetic principles in various engineering and scientific applications.

Concept:

In a magnetic circuit, magnetic flux (\( \phi \)) behaves similarly to electric current in an electrical circuit. In a series magnetic circuit, all the magnetic components (like cores and air gaps) are arranged in series, and the same flux flows through each part, regardless of the varying reluctance of the individual sections.

Key Principle:

Just as current remains the same in a series electrical circuit, magnetic flux remains constant in a series magnetic circuit.

Evaluation of Options:

Option 1: the same – Correct
Same flux flows through all elements in a series magnetic circuit.

Option 2: different –  Incorrect
Flux would differ only in parallel magnetic paths, not in series.

Option 3: zero – Incorrect
Flux exists as long as there is magnetomotive force (mmf).

Option 4: infinite –  Incorrect
Infinite flux is not physically possible.

Concept:

According to Lenz’s Law, the direction of the induced current is always such that it opposes the cause producing it. In a transformer, when current flows in the secondary winding, it generates a magnetic field that opposes the magnetic field of the primary coil.

This opposition is what maintains energy conservation and proper transformer action. The effect of this opposing magnetic field is referred to as a demagnetizing effect.

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.

The correct answer is Ampere-Maxwell's law.

Key Points

• Ampere-Maxwell's law: This is a fundamental law of electromagnetism that describes the relationship between a changing electric field and a magnetic field.
  • It states that a time-varying electric field induces a magnetic field, and conversely, a changing magnetic field induces an electric field.
  • This is the phenomenon you described in your question.

Additional Information

• Kirchhoff's law: These laws describe the behavior of electrical circuits and do not directly relate to the relationship between electric and magnetic fields.
• Faraday's law: This law specifically describes the generation of an electric field due to a changing magnetic field, not vice versa.
• Hertz's law: This law relates to the generation of electromagnetic waves by oscillating charges, not specifically to the relationship between electric and magnetic fields in general.

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:

\({\rm{E\;}} = {\rm{\;N\;}}\left( {\frac{{{\rm{d\Phi }}}}{{{\rm{dt}}}}} \right){\rm{Volts}}\)

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.​

\(v=-N\frac{d\text{ }\!\!\Phi\!\!\text{ }}{dt}\)

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:

\(\rm \vec \nabla \times \vec E = -\frac{{\partial \vec B}}{{\partial t}}\)

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

The correct answer is option 3): 1250 V 

Concept:

The Inductance of the coil is given by

L = \(N \phi \over I\) Henry

EMF . induced E = L\(di \over dt\) V

Calculation:

L = \(1000 ×0.25 × 10^{-3}\over 2\)

= 0.125

E = 0.125× \((10 -0) \over 1 \times 10 ^{-3}\)

(Where current changes from 10A to 0 A)

= 1250 V

Explanation:

The energy density in a magnetic field is given by the formula:

\(\mu={BH\over 2}\)

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

Maxwell equations:

• are an extension of the works of Gauss, Faraday, and Ampere
• help to study the application of both electrostatic and magnetic fields
• can be written in integral form and point form
• need to be modified depending upon the media involved in the problem.

Important Points:

Maxwell’s equation for static electromagnetic fields are as shown:

Explanation:

• Faraday’s first law states that whenever there is a change in the magnetic flux linked with a coil or a conductor, an electromotive force (EMF) is induced in the coil. This law describes the fundamental principle of electromagnetic induction, stating that the magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux.
• In the given statement, it describes the generation of EMF when a conductor or coil moves in a magnetic field, causing a change in the magnetic flux linked with the conductor.
• This change in flux induces an EMF in the conductor, in accordance with Faraday’s first law.
• Hence, the statement aligns with Faraday’s first law of electromagnetic induction

CONCEPT:

Lenz's Law:

• According to this law, the direction of induced emf or current in a circuit is such as to oppose the cause that produces it.
• This law gives the direction of induced emf/induced current.
• This law is based upon the law of conservation of energy.

EXPLANATION:

• 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.