Maxwell's Equations Questions

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Very Short Answers Questions [1 Mark Questions]

Ques. The differential form of Gauss’s law in magnetostatics is

1. div B = 0
2. div B = μJ
3. div B = ρ/ε_o
4. div B = -dB/dT

Ans. The correct answer is a. div B = 0

Explanation: In magnetostatics, Gauss's law states that the surface integration of a magnetic field across a closed surface is zero. It has the differential form: div B = 0.

Ques. Which of the following is the expression for Lorentz force?

1. q (v x B)
2. qE
3. qE + q (v x B)
4. ma + qE

Ans. The correct answer is c. qE + q (v x B)

Explanation: Lorentz force is the force exerted on a particle when it passes through a medium containing both magnetic and electric fields.

Ques. The magnetic field can be produced by

1. Displacement current
2. Conduction current
3. It is produced naturally
4. Both conduction and displacement current

Ans. The correct answer is d. Both conduction and displacement current

Explanation: It was thought that only conduction current could produce a magnetic field. Maxwell later demonstrated that displacement current can also produce a magnetic field. It was known as the Maxwell-Ampere law.

Ques. The velocity of a charged particle to keep moving in the same direction, in a region where electric and magnetic fields are perpendicular to each other, is

1. E/B
2. E/B + qE/B
3. B/E
4. B/E + qB/E

Ans. The correct answer is a. E/B

Explanation: Lorentz force is given by qE + q (v X B).

As we know, the net force on the particle should be zero for it to travel in the same direction.

As a result, qE + q (v X B) = 0.

When we solve for v, we obtain v = E/B.

Ques. The property of a magnetic field to converge electrons is used in microscopes as

1. Magnetic field
2. Magnetic plate
3. Magnetic Lens
4. Magnetic Convergence

Ans. The correct answer is c. Magnetic Lens

Explanation: The property of a magnetic field to converge electrons is used in microscopes as a Magnetic Lens.

Ques. The entire theory of electromagnetic waves is contained in Maxwell’s equations.

1. True
2. False

Ans. The correct answer is a. True

Explanation: Maxwell's equation is a set of four equations that may be used to solve and understand every concept in Maxwell's equation.

Ques. Maxwell’s first equation is based on

1. Faraday’s law
2. Gauss’s law for electrostatic
3. Ampere’s circuital law
4. Gauss’s law for magnetism

Ans. The correct answer is b. Gauss’s law for electrostatic

Explanation: The first equation of Maxwell is based on Gauss' electrostatics law. The density of an electric flux of a closed surface integral is always equal to the charge contained across the surface, according to Gauss law.

Ques. Maxwell’s fourth equation is based on

1. Ampere’s circuital law
2. Ohm’s law
3. Faraday’s law
4. Coulomb’s law

Ans. The correct answer is a. Ampere’s circuital law

Explanation: The fourth equation of Maxwell is based on Ampere's circuital law. Ampere's circuital law is essential for understanding Maxwell's fourth equation.

Ques. In vacuum or free space, what observations are made?

1. ρ = ρ_o, J = 0
2. ρ = ρ_o, J = J_o
3. ρ = 0, J = J_o
4. ρ = 0, J = 0

Ans. The correct answer is d. ρ = 0, J = 0

Explanation: There is no charge or current in a vacuum or free space. Thus, in that situation, ρ and the current density, J, are both equal to zero.

Ques. Which of the following is the expression for the continuity equation?

1. dρ/dt + J = 0
2. dρ/dt + div.J = 0
3. ρ + J = 0
4. ρ + div J = 0

Ans. The correct answer is b. dρ/dt + div.J = 0

Explanation: The continuity equation shows that the sum of the conduction and displacement currents has the property of continuity.

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Short Answers Questions [2 Marks Questions]

Ques. What is the Maxwell equation?

Ans. The four basic equations of electromagnetism are the Maxwell equations, which include Faraday's law of electromagnetic induction, Gauss' law of electricity, Ampere's law of current-carrying conductors, and Gauss' law of magnetism.

Ques. Write the four Maxwell’s equations.

Ans. The four main Maxwell’s equations are

• Maxwell’s first equation (Gauss’s law for electricity)

∇.E = \(\frac{\rho}{\epsilon _0}\)

• Maxwell’s second equation (Gauss’s law for magnetism)

∇.B = 0

• Maxwell’s third equation (Faraday’s law)

∇ x E = – \(\frac{dB}{dt}\)

• Maxwell’s fourth equation (Ampere-Maxwell’s law)

∇ x H = – \(\frac{dD}{dt} +J\)

Ques. What do Maxwell’s equations explain?

Ans. Maxwell's equations describe how magnetic and electric fields are produced by electric charges and currents. It describes how an electric field can produce a magnetic field and vice versa.

Ques. What is the relationship between Ohm’s law and Maxwell’s equation?

Ans. Ohm's law is one of the eight sets of the original Maxwell's equations. The equation F, which defines the connection between an electric current and the electromotive force, generates it.

Ques. What is displacement current in Maxwell’s equation?

Ans. Maxwell's equations contain a component known as displacement current. The rate of change of the electric displacement field (D) is used to define displacement current.

Ques. Is it true that Maxwell’s equations are always correct?

Ans. When quantum mechanical processes are present, Maxwellian electrodynamics fails, and quantum mechanics must replace Newtonian mechanics in that regime. These equations do not "fail" since in quantum mechanics, there is always an equivalent version; only the mechanics differ.

Ques. What is electromagnetic induction?

Ans. The production of an electromotive force across an electrical conductor in a changing magnetic field is known as electromagnetic induction. The phenomenon of electromagnetic induction was discovered in 1831 by Michael Faraday, and James Clerk Maxwell mathematically described it as Faraday's law of induction.

Ques. What is Ampere’s circuital law?

Ans. According to Ampere's circuital law, the closed line integral of a magnetic field vector is always equal to the entire amount of scalar electric field enclosed within the path of any form, which implies the current flowing into the wire is equal to the magnetic field vector.

Ques. What is the physical significance of the Gaussian surface?

Ans. A Gaussian surface is a three-dimensional enclosed surface used to calculate the flux of a vector field (gravitational field, electric field, or magnetic field). The Gaussian surface helps in calculating the intensity of the electric field gently due to the uniformly distributed charge.

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Long Answers Questions [3 Marks Questions]

Ques. If the velocity of a charged particle in a perpendicular electric and magnetic field is 7.27 x 10⁶ m/s and the Electric field is 6 x 10⁶ N/C, what should be the value of the magnetic field for the velocity sector?

Ans. Given

• The velocity of the charged particle, v = 7.27 x 10⁶ m/s
• Electric field, E = 6 x 10⁶ N/C

The velocity of a charged particle in a perpendicular electric and magnetic field is given by

v = E/B

⇒ B = E/v

Where B is the magnetic field.

On substituting the values, we get

B = (6 x 10⁶)/(7.27 x 10⁶)

⇒ B = 0.83 T

Ques. In an electromagnetic wave, the electric field of amplitude 6.2 V/m oscillates with a frequency of 2.4 x 10¹⁰ Hz. Calculate the energy density of the wave.

Ans. Given

• The amplitude of the electric field, E₀ = 6.2 V/m
• Frequency of the electric field, f = 2.4 x 10¹⁰ Hz

According to Maxwell’s equations, energy density is given by

Energy density = ∈₀E₀²

Where

• ∈₀ is the absolute permittivity of the free space = 8.854 x 10^-12 C²N/m²
• E₀ is the amplitude of the electric field

On substituting the values, we get

Energy density = (8.854 x 10^-12) x (6.2)²

⇒ Energy density = 3.4 x 10^-10 J/m³

Ques. What are the limitations of Maxwell's Equations?

Ans. The following are the limitations of Maxwell’s Equations

• Maxwell's equations cannot correctly predict the observations of quantum phenomena such as thermal radiation and the operation of the laser.
• Maxwell's equations have an inherent inconsistency involving the self-force of a charge, which still has no known solution.
• When a charge is accelerated, it radiates energy. This is because the field has mass or inertia associated with it.
• The field mass is in addition to the mass of the particle itself.
• The field must produce an equal but opposite force on the object that is accelerating the charge.

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Very Long Answers Questions [5 Marks Questions]

Ques. What is the physical significance of Maxwell’s equations?

Ans. The following are the physical significance of Maxwell’s equations

• Physical significance of Maxwell's 1^st equation: According to Gauss's law of electrostatics, the total electric flux through any closed surface is 1/∈₀ times the total charge contained by the closed surfaces. This is a steady-state equation since it does not change over time. Divergence of the electric field is positive for positive ρ, and negative for negative ρ. It denotes that ρ is a scalar quantity.
• Physical significance of Maxwell's 2^nd equation: It expresses the magnetostatic Gauss law as ∏. B = 0, implying that isolated magnetic poles or magnetic monopoles cannot exist because they only arise in pairs and there is no source or sink for magnetic lines of forces. It is also time-independent, i.e. a steady-state equation.
• Physical significance of Maxwell's 3^rd equation: It shows that an electric field is formed by a time-varying magnetic flux in line with Faraday's law of electromagnetic induction. This equation is time-dependent.
• Physical significance of Maxwell's 4^th equation: This is a time-dependent equation that describes the modified differential version of Ampere's circular law, which states that a magnetic field is created by the combined impact of conduction and displacement current density.

Ques. Why did Maxwell modify Ampere’s law?

Ans. Maxwell modified Ampere's law to account for the influence of time-varying electric fields. Time-varying electric flux can be used to generate magnetic fields. Ampere's law said,

\(\oint\) \(\vec{B}. \vec{d}\)l = μ₀I

Maxwell discovered that displacement current (Id) and conduction current I had the property of continuity along a closed path in Ampere's law. The displacement current is given by

I_d = ∈₀ \(\frac{d \phi _E}{dt}\)

Maxwell modified Ampere's law to include displacement current (Id)

\(\oint\)\(\vec{B}. \vec{d}\)l = μ₀ \((I + \frac{d \phi _E}{dt})\)

This is known as Ampere-Maxwell law.

Maxwell modified Ampere's law by including the displacement current into the current electric component to satisfy the electric charge continuity equation.

Ques. What are the applications of Maxwell’s equation?

Ans. The following are the applications of Maxwell’s equation

• The static electric field in a vacuum can be calculated using Maxwell's equation.
• It is used for calculating the magnetic field in a vacuum.
• Maxwell's equation is used as a mathematical model for optical, electric, radio, and power-generating technologies, as well as lenses, wireless communication, and radar.
• Maxwell's equations are used to design generators, transformers, and transmission lines, which are essential for generating and distributing electricity.
• Maxwell's equations are used to design antennas and other devices for transmitting and receiving electromagnetic waves, which are used for radio, television, and cellular communications.
• Maxwell's equations are used to study the properties of materials, such as their conductivity and permeability. This knowledge is used to develop new materials with improved properties.

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