Magnetostatics MCQ Quiz - Objective Question with Answer for Magnetostatics - Download Free PDF

Force Acting on Two Parallel Current-Carrying Conductors

Definition: When two parallel current-carrying conductors are placed near each other, they exert a force on each other due to the interaction of their magnetic fields. The direction and nature of this force depend on the direction of the currents in the two conductors. If the currents flow in the same direction, the force is attractive, and if the currents flow in opposite directions, the force is repulsive.

Working Principle:

The phenomenon can be explained using Ampere's law and the Biot-Savart law:

• Each conductor generates a magnetic field around it due to the flow of current. The magnetic field produced by a current-carrying conductor at a distance r from it is given by:

B = (μ₀ × I) / (2π × r)

• Here, B is the magnetic field, μ₀ is the permeability of free space, I is the current, and r is the distance from the conductor.
• The second conductor, placed in the magnetic field of the first conductor, experiences a force due to the magnetic field. This force is given by:

F = I × L × B

• Here, F is the force, L is the length of the conductor, and B is the magnetic field.

Substituting the value of B into the formula for force:

F = (μ₀ × I₁ × I₂ × L) / (2π × r)

• Here, I₁ and I₂ are the currents in the two conductors, and r is the distance between them.

The direction of the force can be determined using the right-hand rule. If the currents in the two conductors are in the same direction, the force is attractive. If the currents are in opposite directions, the force is repulsive.

Correct Option Analysis:

The correct option is:

Option 4: Attractive

When the currents in the two parallel conductors flow in the same direction, the magnetic fields around the conductors interact in such a way that the conductors attract each other. This phenomenon is a direct consequence of the magnetic field interaction and is consistent with the principles of electromagnetism.

The force of attraction can be calculated using the formula:

F = (μ₀ × I₁ × I₂ × L) / (2π × r)

• Here, F is the attractive force per unit length of the conductors.
• μ₀ is the permeability of free space (4π × 10⁻⁷ H/m).
• I₁ and I₂ are the currents in the two conductors.
• L is the length of the conductors.
• r is the distance between the conductors.

This formula highlights that the force is proportional to the product of the currents and inversely proportional to the distance between the conductors.

Additional Information

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

Option 1: Zero

This option is incorrect because the force is not zero. The magnetic fields produced by the currents interact with each other, resulting in a force between the conductors. The force can only be zero if the currents are zero or the conductors are infinitely far apart, which is not the case here.

Option 2: Infinite

This option is incorrect because the force cannot be infinite. The force depends on the currents, the distance between the conductors, and the permeability of free space, all of which are finite quantities under normal circumstances.

Option 3: Repulsive

This option is incorrect because the force is repulsive only when the currents in the two conductors flow in opposite directions. In this case, since the currents are in the same direction, the force is attractive.

Option 5: (No Option Provided)

This option is not valid because it does not contain any relevant information or alternative explanation for the phenomenon.

Conclusion:

The force acting on two parallel current-carrying conductors is attractive if the currents flow in the same direction. This attractive force arises from the interaction of the magnetic fields generated by the currents in the conductors. Understanding this concept is essential for applications in electromagnetism, such as in the design of electrical machines and power transmission systems.

Explanation:

Ohm's Law for Magnetic Circuits:

Definition: Ohm's law for a magnetic circuit states that the magnetomotive force (MMF) is equal to the product of the magnetic flux and the magnetic reluctance. This relationship is analogous to Ohm's law in electrical circuits, where the voltage is equal to the product of current and resistance.

Mathematical Representation:

The equation for Ohm's law in a magnetic circuit is:

MMF = Φ × R

• MMF: Magnetomotive force, measured in ampere-turns (At).
• Φ: Magnetic flux, measured in webers (Wb).
• R: Magnetic reluctance, measured in ampere-turns per weber (At/Wb).

Correct Option Analysis:

The correct option is:

Option 3: Magnetomotive force to magnetic reluctance.

This option correctly represents the relationship defined by Ohm's law for magnetic circuits. The magnetomotive force (MMF) is the driving force that creates magnetic flux in a magnetic circuit. When divided by the magnetic reluctance, it gives the magnetic flux, which is analogous to the current in an electrical circuit. Therefore, MMF is proportional to the product of magnetic flux and magnetic reluctance, aligning with the principles of Ohm's law.

Detailed Explanation:

In magnetic circuits, just as in electrical circuits, there is a driving force, a flow, and opposition to flow. These terms correspond to:

• Driving Force: Magnetomotive force (MMF), analogous to voltage in electrical circuits.
• Flow: Magnetic flux (Φ), analogous to current in electrical circuits.
• Opposition: Magnetic reluctance (R), analogous to resistance in electrical circuits.

When MMF is applied across a magnetic circuit, it generates magnetic flux, which flows through the circuit. The opposition to this flow is provided by the magnetic reluctance, which depends on factors like the material's permeability, cross-sectional area, and length of the magnetic path.

The analogy to Ohm's law in electrical circuits makes it easier to understand and analyze magnetic circuits using similar principles. In this context:

• MMF: Represented as the product of current (I) and the number of turns (N) in the coil, i.e., MMF = N × I.
• Magnetic Flux: The quantity of magnetic field passing through a given area, denoted as Φ.
• Magnetic Reluctance: The opposition to the formation of magnetic flux, calculated using the formula R = l/(μ × A), where:
  • l: Length of the magnetic path.
  • μ: Permeability of the material.
  • A: Cross-sectional area of the magnetic path.

Thus, the correct option, "Magnetomotive force to magnetic reluctance," aligns with the established relationship in magnetic circuits.

Additional Information

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

Option 1: Magnetic reluctance to magnetic flux.

This option is incorrect because it misrepresents the relationship defined by Ohm's law. Magnetic reluctance is the opposition to magnetic flux, and its ratio to magnetic flux does not define MMF. Instead, MMF is the product of flux and reluctance, as established by the law.

Option 2: Magnetic reluctance to magnetomotive force.

This option is incorrect as well because it inverses the relationship. Magnetic reluctance, when divided by MMF, does not yield a meaningful physical quantity in the context of Ohm's law for magnetic circuits. The flux is obtained by dividing MMF by reluctance, not the other way around.

Option 4: Magnetomotive force to permeance.

This option is partially misleading. Permeance is the reciprocal of reluctance and represents the ease with which magnetic flux can be established in a material. While MMF divided by permeance equals magnetic flux, this option is not the most direct representation of Ohm's law for magnetic circuits, which specifically uses reluctance as the term for opposition.

Conclusion:

Understanding the principles of Ohm's law for magnetic circuits is essential for analyzing magnetic systems. The correct option, "Magnetomotive force to magnetic reluctance," accurately describes the fundamental relationship in such circuits, where MMF drives magnetic flux through a material, overcoming the opposition of magnetic reluctance.

Concept:

The magnetic field intensity H due to an infinitely long straight current-carrying conductor at a distance r is given by:

\(H = \frac{I}{2\pi r} \text{ A/m}\)

Given:

• Current I = 10 A
• Distance between wires = 5 m
• Point P is 1 m away from the left wire and 4 m away from the right wire
• Currents are in opposite directions

Calculation:

Magnetic field at P due to the left wire:

H₁ = \(\frac{10}{2\pi \cdot 1} = \frac{10}{2\pi} = \frac{5}{\pi}\)A/m

Magnetic field at P due to the right wire:

H2 = \(\frac{10}{2\pi \cdot 4} = \frac{10}{8\pi} = \frac{5}{4\pi} \text{ A/m}\)

Since currents are in opposite directions, the magnetic fields will add up at point P (in the same direction due to the right-hand rule):

\(H = H_1 + H_2 = \frac{5}{\pi} + \frac{5}{4\pi} = \frac{20 + 5}{4\pi} = \frac{25}{4\pi} \text{ A/m}\)

Convert to match the options given (denominator \(8\pi\)):

\(H = \frac{25}{4\pi} = \frac{50}{8\pi} \text{ A/m}\)

Hence, the correct option is 1

Explanation:

Magnetic Field Intensity (H):

Definition: Magnetic field intensity, denoted as H, is a vector quantity that represents the magnetizing force or the strength of a magnetic field. It measures how intensely a magnetic field is generated by a current-carrying conductor or by magnetic materials. The unit of magnetic field intensity is ampere-turns per meter (AT/m), which indicates the amount of current and the number of turns in a coil per unit length of the magnetic field.

Formula: Magnetic field intensity is mathematically expressed as:

H = (N × I) / L

• H: Magnetic field intensity (AT/m)
• N: Number of turns in the coil
• I: Current in the coil (amperes)
• L: Length of the magnetic path (meters)

Explanation of the Correct Option:

The correct unit for magnetic field intensity H is AT/m, i.e., ampere-turns per meter. This unit is derived from the relationship between the current flowing through a coil, the number of turns in the coil, and the length of the magnetic path.

When a current passes through a coil, it creates a magnetic field around the coil. The magnetic field intensity is directly proportional to the product of the current and the number of turns in the coil and inversely proportional to the length of the magnetic path. This relationship is fundamental to electromagnetism and is used in the design of magnetic circuits in various applications like transformers, motors, and generators.

Applications:

• Magnetic field intensity is a critical parameter in the design of electromagnetic devices like solenoids, inductors, and electromagnets.
• It helps in determining the strength of the magnetic field required for specific applications.
• Used in the analysis of magnetic circuits to evaluate the magnetizing force needed for magnetic materials to reach saturation.

Important Information

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

Option 2: ATm

This option is incorrect because ATm, or ampere-turns multiplied by meter, does not represent the unit of magnetic field intensity. Instead, it could represent a parameter related to the total magnetomotive force (MMF) when multiplied by the length of the magnetic path. Magnetic field intensity is a ratio and specifically involves the division of ampere-turns by meter, not multiplication.

Option 3: Am

This option is incorrect as well. Am, or ampere-meter, does not represent the unit of magnetic field intensity. This unit might be used to describe certain other electromagnetic properties, but it is not relevant in the context of magnetic field intensity. Magnetic field intensity requires the inclusion of the number of turns in the coil, making AT/m the correct unit.

Option 4: AT/m²

This option is also incorrect. AT/m², or ampere-turns per square meter, might suggest a quantity related to the density of magnetizing force over an area, but this is not the standard unit used for magnetic field intensity. Magnetic field intensity is measured in terms of ampere-turns per meter (AT/m), which is a linear measurement.

Conclusion:

Magnetic field intensity is a fundamental concept in electromagnetism, essential for understanding and analyzing magnetic circuits. The correct unit for magnetic field intensity is AT/m (ampere-turns per meter). By evaluating the other options, we confirm that none of them accurately represents the unit for magnetic field intensity, which is critical in designing and analyzing electromagnetic systems.

Explanation:

Ampere's Circuital Law for Time-Varying Fields

Definition: Ampere's Circuital Law is a fundamental relationship in electromagnetics that relates the magnetic field circulating around a closed path to the electric current passing through the surface enclosed by that path. For time-varying fields, the law is modified to include the contribution of displacement current, which accounts for the changing electric field.

Mathematical Expression:

For time-varying fields, Ampere's Circuital Law is expressed as:

\[\nabla \times \overline{\mathrm{H}} = \overline{\mathrm{J}_{\mathrm{C}}} + \overline{\mathrm{J}_{\mathrm{D}}}\]

where:

• \(\nabla \times \overline{\mathrm{H}}\): The curl of the magnetic field intensity.
• \(\overline{\mathrm{J}_{\mathrm{C}}}\): The conduction current density.
• \(\overline{\mathrm{J}_{\mathrm{D}}}\): The displacement current density.

Displacement Current: The concept of displacement current was introduced by James Clerk Maxwell to address the inconsistency in Ampere's Law for time-varying fields. It is given by:

\[\overline{\mathrm{J}_{\mathrm{D}}} = \epsilon_0 \frac{\partial \overline{\mathrm{E}}}{\partial t}\]

where:

• \(\epsilon_0\): The permittivity of free space.
• \(\frac{\partial \overline{\mathrm{E}}}{\partial t}\): The rate of change of the electric field.

The term displacement current is not a physical current but an apparent current that arises from the time-varying electric field.

Correct Option Analysis:

The correct expression for Ampere's Circuital Law in the presence of time-varying fields is:

\[\nabla \times \overline{\mathrm{H}} = \overline{\mathrm{J}_{\mathrm{C}}} + \overline{\mathrm{J}_{\mathrm{D}}}\]

Thus, the correct answer is Option 2.

Additional Information

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

Option 1: \(\nabla \times \overline{\mathrm{H}} = \overline{\mathrm{J}_{\mathrm{C}}} - \overline{\mathrm{J}_{\mathrm{D}}}\)

This option incorrectly subtracts the displacement current density \(\overline{\mathrm{J}_{\mathrm{D}}}\) from the conduction current density \(\overline{\mathrm{J}_{\mathrm{C}}}\). In Ampere's Circuital Law for time-varying fields, these two terms are additive because both contribute to the curl of the magnetic field intensity.

Option 3: \(\nabla \times \overline{\mathrm{H}} = \mathrm{J}_{\mathrm{C}} + \mathrm{J}_{\mathrm{D}}\)

This option has the correct additive relationship but uses scalar current densities (\(\mathrm{J}_{\mathrm{C}}\) and \(\mathrm{J}_{\mathrm{D}}\)) instead of vector current densities (\(\overline{\mathrm{J}_{\mathrm{C}}}\) and \(\overline{\mathrm{J}_{\mathrm{D}}}\)). Since current density is a vector quantity, this representation is incomplete and, therefore, incorrect.

Option 4: \(\nabla \times \overline{\mathrm{H}} = \mathrm{J}_{\mathrm{C}} - \mathrm{J}_{\mathrm{D}}\)

This option is incorrect for two reasons: it subtracts the displacement current density instead of adding it, and it uses scalar current densities instead of vector current densities. Both aspects deviate from the proper formulation of Ampere's Circuital Law for time-varying fields.

Conclusion:

Ampere's Circuital Law is a cornerstone of electromagnetics, especially in the context of Maxwell's equations. For time-varying fields, the inclusion of displacement current ensures the consistency of the law and its applicability to scenarios involving changing electric and magnetic fields. The correct mathematical expression:

\[\nabla \times \overline{\mathrm{H}} = \overline{\mathrm{J}_{\mathrm{C}}} + \overline{\mathrm{J}_{\mathrm{D}}}\]

highlights the contributions of both conduction current and displacement current to the magnetic field. This understanding is critical for analyzing electromagnetic wave propagation, antenna design, and other advanced topics in electromagnetics.

CONCEPT:

• Magnetic field strength or magnetic field induction or flux density of the magnetic field is equal to the force experienced by a unit positive charge moving with unit velocity in a direction perpendicular to the magnetic field.
  • The SI unit of the magnetic field (B) is weber/meter2 (Wbm-2) or tesla.
• The CGS unit of B is gauss.

​1 gauss = 10-4 tesla.

EXPLANATION:

• From the above explanation, we can see that the relation between tesla and Weber/m2 is given by:

1 tesla = 1 Weber/m2

CONCEPT:

• Fleming's Left-hand rule gives the force experienced by a charged particle moving in a magnetic field or a current-carrying wire placed in a magnetic field.
• It states that "stretch the thumb, the forefinger, and the central finger of the left hand so that they are mutually perpendicular to each other.
• If the forefinger points in the direction of the magnetic field, the central finger points in the direction of motion of charge, then the thumb points in the direction of force experienced by positively charged particles."

EXPLANATION:

• According to question

1. Forefinger (Index finger): Represents the direction of the magnetic field (magnetic flux). Therefore option 3 is correct.
2. Middle finger: Represents the direction of motion of charge (current).
3. The thumb : Represents the direction of force or motion experienced by positively charged particles.

Concept:

Magnetic Field Strength (H): the amount of magnetizing force required to create a certain field density in certain magnetic material per unit length.

\(H=\frac{NI}{L}\)

The intensity of Magnetization (I): It is induced pole strength developed per unit area inside the magnetic material.

The net Magnetic Field Density (Bnet) inside the magnetic material is due to:

• Internal factor (I)
• External factor (H)
   

∴ Bnet ∝  (H + I)

Bnet = μ0(H + I) …. (1)

Where μ0 is absolute permeability.

Note: More external factor (H) causes more internal factor (I).

∴ I ∝  H

I = KH …. (2)

And K is the susceptibility of magnetic material.

From equation (1) and equation (2):

Bnet = μ0(H + KH)

Bnet = μ0H(1 + K) …. (3)

Dividing equation (3) by H on both side

\(\frac{{{B_{net}}}}{H} = \frac{{{μ _0}H\left( {1 + K} \right)}}{H} \)

or, μ0μr = μ0(1 + K)

∴ μ_r = (1 + K) .... (4)

From equation (3) and (4)

Bnet = μ0μ_rH

Calculation:

Given Magnetic Circuit,

N = 100
I = 5 A
L = 2πr = 2π × 5 × 10^-2 m

From above concept,

\(H=\frac{NI}{L}=\frac{100× 5}{10π × 10^{-2}}=\frac{5000}{π}\)

We know that,

Bnet = μ0μrH

And, μ_r = 1000

\(B_{net}=4\pi \times 10^{-7}\times 1000\times \frac{5000}{\pi}=2\ T\)

Concept:

The magnetic field intensity (H) of a circular coil is given by

\(H = \frac{I}{{2 R}}\)

Where I is the current flow through the coil

R is the radius of the circular coil

Calculation:

Given that, Current (I) = 2 A                                             

Diameter = 1 m

Radius (R) = 0.5 m

Magnetic field intensity \(H = \frac{2}{{2 \times 0.5}} = 2\;A/m\)

Common Mistake:

The magnetic field intensity (H) of a circular coil is given by \(H = \frac{I}{{2 R}}\)

The force between two current-carrying parallel conductors:

• Two current-carrying conductors attract each other when the current is in the same direction and repel each other when the currents are in the opposite direction
• Force per unit length on conductor

\(\frac{F}{l}=~\frac{{{\mu }_{0}}}{2\pi }\frac{{{i}_{1}}{{i}_{2}}}{d}\)

Faraday’s first law of electromagnetic induction 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 states that the magnitude of emf 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.

The magnetizing field strength due to long straight circular conductor is given by

\(H = \;\frac{I}{{2\pi r}}\;AT/m\)

Where, H = Magnetizing force (AT/m)

I = Current flowing in a conductor (A)

r = Distance between current carrying conductor and the point (m)

also B = μ₀ H

Where, B = Magnetic flux density (Wb/m²)

μ₀ = Absolute permeability = 4π × 10^-7 H/m

Calculation:

Given:

r = 5 cm = 5 × 10^-2 m

I = 250 A

\(B = {\mu _0}H = {\mu _0}\frac{I}{{2\pi r}} = 4\pi \times {10^{ - 7}} \times \;\frac{{250}}{{2\pi\times 5 \times {{10}^{ - 2}}}} = {10^{ - 3}}\)

∴ B = 10^-3 Wb/m²

^{Important Points}

 Magnetic flux density of coil is given by ^{\(B = \;\frac{{{\mu _o}NI}}{{2R}}\)}  T

Magnetic flux density of solenoid is given by \(B = \;\frac{{{\mu _0}\;NI}}{l}\)  T

Magnetic flux density of long straight wire \(B = \;\frac{{{\mu _0}I}}{{2\pi r}}\) T

The magnetizing field strength due to long straight circular conductor is given by

\(H = \;\frac{I}{{2\pi r}}\;AT/m\)

Where, H = Magnetizing force (AT/m)

I = Current flowing in a conductor (A)

r = Distance between current carrying conductor and the point (m)

also B = μ0 H

Where, B = Magnetic flux density (Wb/m2)

μ0 = Absolute permeability = 4π × 10-7 H/m

Calculation:

Given:

r = 5 cm = 5 × 10-2 m

I = 250 A

\(H = \frac{I}{{2\pi r}} = \frac{{250}}{{2\pi\times 5 \times {{10}^{ - 2}}}} = \frac{2500}{\pi}\)

Important Points

 Magnetic flux density of coil is given by \(B = \;\frac{{{\mu _o}NI}}{{2R}}\)  T

Magnetic flux density of solenoid is given by \(B = \;\frac{{{\mu _0}\;NI}}{l}\)  T

Magnetic flux density of long straight wire \(B = \;\frac{{{\mu _0}I}}{{2\pi r}}\) T

Magnetic Field at the center of a Circular Current carrying coil:

Consider a current-carrying circular loop having its center at O carrying current i as shown below,

If dl is a small element at a distance r then, the magnetic field intensity on that point can be written using Biot-Savart’s law.

\(|\vec{dB}|=\frac{\mu_0i}{4π}\frac{idlsinθ}{r^2}\) .... (1)

Since, the loop is circular,

θ = 90° → sin θ = 1

Hence, equation (1) becomes,

\(|\vec{dB}|=\frac{\mu_0i}{4π}\frac{idl}{r^2}\)

If the circular loop is consisting of numbers of small element dl, then we will get the magnetic intensity all over the loop.

Hence to get the total field we must sum up that is integrate the magnetic field all over the field.

\(B=∫{\vec{dB}}\)

\(B=∫|\vec{dB}|=∫\frac{\mu_0i}{4π}\frac{idl}{r^2}\)

\(B=\frac{\mu_0i}{4π r^2}∫ dl\)

∫dl = l = 2πr

Hence,

\(B=\frac{\mu_oi}{2r}\)

Now, we have to find magnetic field intensity or magnetizing force (H),

And, H = \(\frac{B}{\mu_0}=\frac{i}{2r}\)

Calculation:

Given the length of the wire is 2 m and it bends to the circle, hence the circumference of the circle will be 2 m.

Now, We have, 2πr = 2 m, r = 1/π m
Magnetizing force at the centre of the circle, H = \(\frac{i}{2r}=\frac{50}{2\times1/π}=25π\)
H = 25π = 78.54 AT/m

CONCEPT: 

The magnetic field at point P due to a straight conductor is given by:

B = \(μ_0 I \over{2\pi d}\)

where B is the magnetic field at point P, μ0 permeability of the medium I is the current in the in the wire and d is the distance from the wire to that point.

EXPLANATION:

At distance r from an infinitely long straight current carrying conductor, magnetic field is:

B = \(μ_0 I \over{2\pi r}\)

where B is the magnetic field, μ0 permeability of the medium I is the current in the wire, and r is the distance from the wire to that point.

So B α 1/r

Important Points

 Magnetic flux density of coil is given by \(B = \;\frac{{{\mu _o}NI}}{{2R}}\)  T

Magnetic flux density of solenoid is given by \(B = \;\frac{{{\mu _0}\;NI}}{l}\)  T

Magnetic flux density of long straight wire \(B = \;\frac{{{\mu _0}I}}{{2\pi r}}\) T