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

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.

Concept

The average EMF induced in the coil is given by:

\(E=-N{Δ ϕ \over Δ t}\)

The magnetic flux is given by:

\(ϕ = {NI\over R}\)

where, E = EMF

N = No. of turns

Δϕ = Change in flux

Δt = Change in time

I = Current

R = Reluctance

Calculation

Given, N = 100

I = 1 A

R = 1000 AT/Wb

When the current reverses, the flux changes from $+0.1$ Wb to $-0.1$ Wb.

The change in flux is given by:

Δϕ = ϕ_final - ϕ_initial

Δϕ = (−0.1) − (0.1) = −0.2 Wb

\(E=-(100)× {-0.2 \over 10× 10^{-3}}\)

E = 100 × 20 × 10^-3 V

E = 2 V

Explanation:

Proper Application of the Divergence Theorem to Coulomb's Law Results in Gauss's Law

Introduction:

The relationship between Coulomb's law and Gauss's law is a fundamental concept in electromagnetism. Coulomb's law describes the force between two point charges, whereas Gauss's law relates the electric flux through a closed surface to the charge enclosed by that surface. The correct mathematical tool to transition from Coulomb's law to Gauss's law is the Divergence Theorem, also known as Gauss's Theorem.

Coulomb's Law:

Coulomb's law states that the electric force \( \mathbf{F} \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:

\(\mathbf{F} = k_e \frac{q_1 q_2}{r^2} \hat{\mathbf{r}}\)

where \( k_e \) is Coulomb's constant and \( \hat{\mathbf{r}} \) is the unit vector in the direction of the force.

Electric Field:

The electric field \( \mathbf{E} \) due to a point charge \( q \) at a distance \( r \) is given by:

\(\mathbf{E} = k_e \frac{q}{r^2} \hat{\mathbf{r}}\)

For multiple charges, the electric field is the vector sum of the fields due to each charge.

Gauss's Law:

Gauss's law states that the total electric flux \( \Phi_E \) through a closed surface \( S \) is equal to the charge enclosed \( Q_{\text{enc}} \) divided by the permittivity of free space \( \epsilon_0 \):

\(\Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0}\)

Divergence Theorem:

The Divergence Theorem relates the flux of a vector field through a closed surface to the volume integral of the divergence of the field. Mathematically, it is expressed as:

\(\oint_S \mathbf{E} \cdot d\mathbf{A} = \int_V (\nabla \cdot \mathbf{E}) dV\)

where \( \mathbf{E} \) is the electric field, \( S \) is the closed surface, and \( V \) is the volume enclosed by \( S \).

Derivation of Gauss's Law from Coulomb's Law using the Divergence Theorem:

To derive Gauss's law from Coulomb's law, we start with the expression for the electric field due to a point charge:

\(\mathbf{E} = k_e \frac{q}{r^2} \hat{\mathbf{r}}\)

The divergence of the electric field for a point charge can be found using the Dirac delta function \( \delta(\mathbf{r}) \):

\(\nabla \cdot \mathbf{E} = \nabla \cdot \left( k_e \frac{q}{r^2} \hat{\mathbf{r}} \right) = \frac{q}{\epsilon_0} \delta(\mathbf{r})\)

Applying the Divergence Theorem:

\(\oint_S \mathbf{E} \cdot d\mathbf{A} = \int_V (\nabla \cdot \mathbf{E}) dV = \int_V \frac{q}{\epsilon_0} \delta(\mathbf{r}) dV = \frac{q}{\epsilon_0}\)

This shows that the electric flux through a closed surface \( S \) is proportional to the charge enclosed by the surface, which is Gauss's law.

Conclusion:

The Divergence Theorem is the key mathematical tool that allows the transition from Coulomb's law to Gauss's law. By applying the Divergence Theorem to the electric field derived from Coulomb's law, we obtain the integral form of Gauss's law, which relates the electric flux through a closed surface to the charge enclosed within that surface.

Analysis of Other Options:

Option 1: Stokes's Theorem

Stokes's Theorem relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary of the surface. It is given by:

\(\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{A}\)

Stokes's Theorem is not directly related to the derivation of Gauss's law from Coulomb's law as it deals with the curl of a vector field and line integrals, rather than the divergence and flux through a closed surface.

Option 3: Helmholtz's Theorem

Helmholtz's Theorem states that a vector field is uniquely determined by its divergence and curl, given appropriate boundary conditions. While this theorem is important in vector calculus, it does not directly relate to the derivation of Gauss's law from Coulomb's law.

Option 4: Uniqueness Theorem

The Uniqueness Theorem in electrostatics states that the solution to Poisson's or Laplace's equation for the electric potential is unique if the boundary conditions are specified. This theorem ensures that the electric field and potential are uniquely determined by the charge distribution and boundary conditions, but it is not directly involved in deriving Gauss's law from Coulomb's law.

Concept:

• Maxwell's Equations:
• Maxwell's equations describe the behavior of electric and magnetic fields. Some of these equations are valid only under time-varying conditions, while others are valid for both static and dynamic conditions.
• The equation that is valid for time-varying conditions and not for static conditions is:
  • ∮ E · dl = - dΦₛ / dt, where:
    • Φₛ: Electric flux
    • This equation represents Faraday's law of induction, which states that a changing magnetic field produces a circulating electric field. It is only valid for time-varying conditions.

Calculation:

For time-varying conditions, the correct Maxwell's equation is:

∮ E · dl = - dΦₛ / dt

∴ The correct equation is given by option 3.

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.