Electric Field in Material MCQ Quiz - Objective Question with Answer for Electric Field in Material - Download Free PDF

Explanation:

Boundary Conditions for Time-Varying Fields

Definition: Boundary conditions in electromagnetics describe how electromagnetic field quantities behave at the interface between two different media. For time-varying fields, these boundary conditions are derived from Maxwell's equations and are essential for solving electromagnetic problems involving interfaces.

Correct Option:

The correct option is:

Option 3: The normal component of magnetic flux density is continuous at the boundary.

Explanation of Correct Option:

The boundary condition on the magnetic flux density (B) arises from Gauss's law for magnetism, which states:

∇ • B = 0

This implies that the net magnetic flux through any closed surface is zero, meaning magnetic monopoles do not exist. At the boundary between two media, the integral form of Gauss's law for magnetism can be expressed as:

∮ B • dA = 0

When applied to a small Gaussian pillbox that straddles the boundary, the contributions to the flux integral come from the two faces of the pillbox (one in medium 1 and the other in medium 2). The result is:

B₁ₙ = B₂ₙ

Here, B₁ₙ and B₂ₙ are the normal components of the magnetic flux density in the two media. This shows that the normal component of B is continuous across the boundary, regardless of the properties of the media. This condition is fundamental and applies to all interfaces, making Option 3 correct.

Important Note: While the magnetic flux density's normal component is continuous, the tangential component of B may be discontinuous if there is a surface current at the boundary.

Additional Information:

The continuity of the normal component of magnetic flux density ensures that there are no magnetic monopoles, aligning with Maxwell's equations' fundamental principles. This boundary condition is vital for solving problems involving magnetic fields in different media, such as in transformers, inductors, and waveguides.

Analysis of Other Options

To ensure a comprehensive understanding, let’s evaluate the other options:

Option 1: The tangential component of magnetic field intensity is discontinuous across the surface except for a perfect conductor.

This statement is incorrect. The tangential component of the magnetic field intensity (H) is continuous across the boundary unless there is a surface current density (K) present. If a surface current exists, the tangential components of H in the two media are related by:

(H₂ - H₁)ₜ = K

For a perfect conductor, the tangential component of H at the surface is zero, but this condition applies only to perfect conductors, not general boundaries. Thus, Option 1 is not universally correct.

Option 2: The normal component of electric flux density is discontinuous at the boundary if the surface charge density is zero.

This statement is incorrect. The boundary condition for the normal component of the electric flux density (D) is given by:

D₂ₙ - D₁ₙ = ρₛ

Here, ρₛ is the surface charge density at the boundary. If ρₛ is zero, the normal component of D is continuous across the boundary. The statement in Option 2 contradicts this condition by asserting discontinuity even when the surface charge density is zero.

Option 4: The tangential component of electric field intensity is discontinuous at the surface.

This statement is incorrect. The tangential component of the electric field intensity (E) is continuous across the boundary unless there is a time-varying magnetic field present. The boundary condition for the tangential component of E is derived from Faraday's law:

∇ × E = -∂B/∂t

In the absence of time-varying magnetic fields, the tangential component of E is continuous. However, in the presence of time-varying magnetic fields, the tangential component may vary, but this is a specific case and not a general rule. Thus, Option 4 is not universally correct.

Conclusion:

The correct boundary condition for time-varying fields, as described, is that the normal component of magnetic flux density (B) is continuous at the boundary, making Option 3 correct. This condition is derived from Gauss's law for magnetism and is fundamental to the behavior of magnetic fields in different media. The analysis of other options highlights common misconceptions and emphasizes the importance of understanding Maxwell's equations and their implications for boundary conditions.

Calculation:

The electric field in the region between the two plates is given by E = X / d.

The proton moves at 45° to the vertical if the net acceleration is along that direction. This happens when the electric force qE equals the gravitational force mg.

So, q(X/d) = mg

⇒ X = (mgd) / q

⇒ X = (1.67 × 10⁻²⁷ kg × 9.8 m/s² × 1 × 10⁻² m) / (1.6 × 10⁻¹⁹ C)

⇒ X ≈ 10⁻⁹ V

Answer: (C) 10⁻⁹ V

Ans. (3) Sol.

Given:

V = 3 volts

R = 1.5 Ω

e = 1.6 × 10⁻¹⁹ C

t = 1 s

Now, Current I = V / R = 3 / 1.5 = 2 A

Also, I = Q / t = (n × e) / t

So, n = (I × t) / e = (2 × 1) / (1.6 × 10⁻¹⁹) = 1.25 × 10¹⁹

Concept:

The number of electrons transferred when a charge is rubbed or moved can be calculated using the formula:

Q = n × e

Where:

• Q = total charge transferred
• n = number of electrons transferred
• e = elementary charge = 1.6 × 10^-19 C

Rearranging the formula to find the number of electrons:

n = Q / e

Calculation:

Given:

• Q = 3.52 × 10^-7 C
• e = 1.6 × 10^-19 C

n = (3.52 × 10^-7) / (1.6 × 10^-19) = 2.2 × 10¹² electrons

∴ The number of electrons transferred is 2.2 × 10¹².

Option 2 is correct, i.e. Conductance.

Conductance:

• The degree to which an object conducts electricity, calculated as the ratio of the current which flows to the potential difference present.
• This is the reciprocal of the resistance and is measured in siemens or mhos.

Note:

Conductivity:

• Electrical conductivity is the measure of the amount of electrical current a material can carry or its ability to carry a current.
• It is also known as specific conductance.
• It is an intrinsic property of a material.
• It is denoted by the symbol σ and has SI units of Siemens per meter (S/m).

Resistivity (ρ):

• Resistivity is the resistance per unit length and cross-sectional area.
• It is the property of the material that opposes the flow of charge or the flow of electric current.
• The unit of resistivity is ohm meter.
• A material with high resistivity means it has got high resistance and will resist the flow of electrons.
• A material with low resistivity means it has low resistance and thus the electrons flow smoothly through the material.
• It depends on the material of the conductor but not on its dimensions. ρ is called resistivity.

Resistance:

• It is defined as the hurdles in the path of flow of current.
• The SI unit of resistance is the ohm.
• It is denoted by the symbol Ω.
• It depends on the material of the conductor and the dimensions of the conductor.
• In other words, we can say it is the ratio of voltage to the current.
• R= (rho×length)/area
• Note: rho=Resistivity.

The correct answer is Free Electrons.

Key Points

• Current carriers in solid conductors are Free Electrons.
  • In solid conductors (e.g. metals), there are a large number of free electrons.
  • When an electric field (i.e. PD) is applied to the conductor, the free electrons start drifting in a particular direction to constitute that current.

Additional Information

• Some liquids are conductors of electricity.
  • A Conducting liquid is called an electrolyte (e.g. solution of CuSO₄).
  • In conducting liquids, ions (positive and negative) are the current carriers.
• Under ordinary conditions, gases are insulators.
  • However, when a gas under low pressure is subjected to a high electric field (i.e. high p.d.), Ionisation of in gases molecules takes place, i.e. electrons and positive ions are formed.
• Hence, current carriers in gases are free electrons and positive ions.

Lightning:

The visible discharge of electricity occurs when a region of a cloud acquires an excess electrical charge (either positive or negative) that is sufficient to break down the resistance of air.

Current is a measure of the flow of electric charge over time.

It is defined as   

where q = Charge

t = Time

I = Current

Calculation:

Given:

q = 25 C

I = 2.5 kA

t = 10 milliseconds

Concept:

The magnitude of the force between the conductor is given by

Where;

I_1,2 → current carried by the conductors 

l → Length

d→ Distance between conductors

Calculation:

Given;

I1,2 → current carried by the conductors = 20 A

d→ Distance between conductors = 20 cm = 0.2 m

Then;

The magnitude of the force per meter between the conductor is given by;

Additional Information

Concept of Current Density:

Ohms’ law: At constant temperature, the current through a resistance is directly proportional to the potential difference across the resistance.

V = R I

Where V is the potential difference, R is resistance and I is current flowing

Current density (J): The electric current per unit area is called current density.

Conductivity (σ): The property of a conductor due to which the current flows through it is called the conductivity of that conductor.

Where ρ is resistivity = 1/σ, l is the length and A is the area of the conductor

As, V = R I 

σ = J/E

Superconductivity: It is the state of the material in which its resistance becomes zero as well as it behaves as perfect diamagnetic when its temperature is reduced below transition temperature Tc and the magnetic field is less than the critical magnetic field Hc.

The current density of the superconductor depends on critical magnetic field strength and temperature.

Superconductors are divided into two types based on magnetic properties:

1) Type-I or Ideal or soft superconductor

2) Type-I or Hard superconductor.

Type-I Superconductor:

A superconducting state is completely diamagnetic up to a certain critical magnetic field (Hc). Above Hc, the superconducting state changes it’s behavior and undergoes to normal state abruptly.

This is explained with the help of the following curve:

Type-II Superconductor:

• It loses their superconductivity gradually because they are made-up by a combination of Hard metal and alloys that show different magnetization behavior.
• It shows superconductivity up to a lower critical magnetic field Hc1 and beyond this, the magnetization changes gradually and reaches zero at upper critical magnetic field Hc2 and due to this, it is useful in the preparation of high-field electromagnets.

Important Points:

Characteristics of superconductors in comparison to a normal metal is as shown:

Concept:

Current passing through the conductor, I = q/t = ne / t

Where

n = number of electrons 

e = charge of electron

t = time duration

Calculation:

q = e = 1.62 × 10-19 

t = 1 sec

⇒ I = n × 1.62 × 10-19

⇒ 8 = n × 1.62 × 10-19

⇒ n = 4.93 × 1019 ≈ 5 × 1019

Concept:

The boundary condition for electrostatic fields are defined as:

E1t = E2t (Tangential components are equal across the boundary surface)

Also, the normal component satisfies the following relation:

D1n – D2n = ρs

For a charge-free boundary, ρs = 0, the above expression becomes:

D1n – D2n = 0

D1n = D2n

Calculation:

As Tangential components are equal across the boundary surface

E₁t = E2t  

E1t = 10 Vm-1

so tangential component of the electric field on the other side is:

E2t = 10 Vm-1

Explanation

Conduction current density:

• It is defined as the amount of current or charges that flow through the conduction surface within a time 't'.
• The current carried by conductors due to the flow of charges is called conduction current.
• The conduction current follows Ohm's Law
• It is given by: J_C = σE 

where J_C = Conduction current density

σ = Conductivity

E = Electric field

Displacement current density:

• It is defined as the rate of change of the electric displacement field with respect to time 't'.
• The current due to changing electric field is called displacement current.
• The displacement current does not follow Ohm's Law.
• It is given by: J_D = 

where J_D = Displacement current density

Superconductivity: It is the state of the material in which its resistance becomes zero as well as it behaves as perfect diamagnetic when its temperature is reduced below transition temperature Tc and the magnetic field is less than the critical magnetic field Hc.

Superconductors ripple magnetic field, hence they are creating regions free from magnetic field. 

The current density of the superconductor depends on critical magnetic field strength and temperature.

Superconductors are divided into two types based on magnetic properties:

1) Type-I or Ideal or soft superconductor

2) Type-I or Hard superconductor.

Type-I Superconductor:

A superconducting state is completely diamagnetic up to a certain critical magnetic field (Hc). Above Hc, the superconducting state changes it’s behavior and undergoes to normal state abruptly.

This is explained with the help of the following curve:

Type-II Superconductor:

• It loses their superconductivity gradually because they are made-up by a combination of Hard metal and alloys that show different magnetization behavior.
• It shows superconductivity up to a lower critical magnetic field Hc1 and beyond this, the magnetization changes gradually and reaches zero at upper critical magnetic field Hc2 and due to this, it is useful in the preparation of high-field electromagnets.

Important Points:

Characteristics of superconductors in comparison to a normal metal is as shown: