Magnetic Flux Density MCQ Quiz - Objective Question with Answer for Magnetic Flux Density - Download Free PDF

Explanation:

In a Magnetic Circuit:

Definition: A magnetic circuit is a path followed by magnetic flux. It is analogous to an electrical circuit but uses magnetic fields instead of electric currents. The primary components of a magnetic circuit include magnetic flux (Φ), magnetomotive force (MMF), and reluctance (R).

Working Principle: The magnetomotive force (MMF) in a magnetic circuit is created by a current passing through a coil of wire, creating a magnetic field. This MMF drives the magnetic flux through the magnetic circuit. The relationship between MMF, magnetic flux, and reluctance is given by:

MMF = Φ × R

where:

• MMF (magnetomotive force) is measured in Ampere-Turns (A-t).
• Φ (magnetic flux) is measured in Webers (Wb).
• R (reluctance) is measured in Ampere-Turns per Weber (A-t/Wb).

Reluctance: Reluctance is the opposition to the creation of magnetic flux in the magnetic circuit. It is analogous to resistance in an electrical circuit. The reluctance depends on the length (l) and cross-sectional area (A) of the magnetic path, as well as the material's permeability (μ), and is given by:

R = l / (μ × A)

where:

• l is the length of the magnetic path.
• μ is the permeability of the material.
• A is the cross-sectional area of the magnetic path.

Correct Option Analysis:

The correct option is:

Option 3: The magnetic flux will decrease.

Explanation: According to the relationship MMF = Φ × R, if the reluctance (R) increases and the magnetomotive force (MMF) remains constant, the magnetic flux (Φ) must decrease. This is because the reluctance provides greater opposition to the magnetic flux, thereby reducing its magnitude. An increase in reluctance means that it is harder for the magnetic flux to pass through the magnetic circuit, resulting in a decrease in the magnetic flux.

Important Information

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

Option 1: The magnetomotive force (MMF) will increase.

This option is incorrect because the magnetomotive force (MMF) is a function of the current and the number of turns in the coil (MMF = N × I). An increase in reluctance does not inherently increase the MMF; instead, it affects the magnetic flux (Φ) for a given MMF.

Option 2: The magnetic flux will increase.

This option is incorrect because, as explained, an increase in reluctance leads to a decrease in magnetic flux, not an increase. The opposition to the magnetic flux becomes greater, reducing the amount of flux in the circuit.

Option 4: The resistance to magnetic flux will decrease.

This option is incorrect because reluctance is the magnetic equivalent of resistance. If the reluctance increases, the opposition (or resistance) to the magnetic flux increases, not decreases.

Conclusion:

Understanding the relationship between MMF, magnetic flux, and reluctance is crucial in analyzing magnetic circuits. When the reluctance of a path in a magnetic circuit increases, the magnetic flux decreases if the MMF remains constant. This fundamental principle helps in the design and analysis of magnetic circuits in various electrical and electronic applications.

Explanation:

Magnetic Circuit and Reluctance

A magnetic circuit is a path followed by magnetic flux. It consists of materials with high magnetic permeability that guide the magnetic flux. Reluctance is the opposition that a magnetic circuit presents to the magnetic flux, analogous to electrical resistance in an electrical circuit. It is denoted by the symbol ℜ and is measured in Ampere-Turns per Weber (At/Wb).

Correct Option Explanation:

When the reluctance of one path in a series magnetic circuit increases, the total reluctance of the circuit also increases. According to the relationship Φ = MMF / ℜ, an increase in total reluctance (ℜ) results in a decrease in the total magnetic flux (Φ) if the MMF remains constant. This is because the magnetic flux is inversely proportional to the reluctance.

To understand this better, consider a series magnetic circuit with a constant MMF. If the reluctance of one section of the circuit increases, the opposition to the magnetic flux increases. As a result, the circuit's ability to carry magnetic flux diminishes, leading to a reduction in the total magnetic flux. This explains why the correct option is:

Option 4: The total flux will decrease.

 Additional Information

Let's analyze why the other options are incorrect:

• Option 1: The total flux will increase. This option is incorrect because an increase in reluctance means an increase in opposition to the magnetic flux. Therefore, the total flux cannot increase.
• Option 2: The total flux will remain unchanged. This option is incorrect because an increase in reluctance directly affects the total magnetic flux, causing it to decrease if the MMF is constant.
• Option 3: The flux will divide equally between paths. This option is incorrect because, in a series magnetic circuit, the flux does not divide between paths. The total flux is the same throughout the series circuit, and an increase in reluctance in one part affects the entire circuit's flux.

The correct understanding of magnetic circuits and reluctance is crucial for designing efficient magnetic systems. By ensuring that the reluctance is minimized, the total magnetic flux can be maximized, leading to better performance of magnetic devices such as transformers, inductors, and magnetic actuators.

Ans.(4)

Sol.

A. Electromagnetic induction is a phenomenon in which a changing magnetic field across a wire loop produces an induced emf. When a magnet and a coil move relative to one another, magnetic flux changes, and an electromotive force is generated in the coil.

B. Magnetic flux is the scalar product (dot product) of the magnetic field vector (B) and elementary area vector (dA). Thus, it is a scalar quantity.

C. 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 the number of turns in the coil and flux associated with the coil

D. The working of magnetic braking of trains is based on eddy current

E. Lenz's law states that the polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.

Concept:

Magnetic flux (\(\Phi\)) through a surface is given by:

\(\Phi = \vec{B} \cdot \vec{A} = B A \cos \theta\)

Where:

• \(\vec{B}\): Magnetic field strength.
• A: Area of the surface.
• \(\theta\): Angle between the magnetic field and the normal to the surface.

For a closed surface, the total flux entering the surface must equal the total flux leaving the surface.

Calculation:

Given:

• \(B = 2 \, \text{mT} = 2 \times 10^{-3} \, \text{T}\)
• Radius of hemisphere \(R = \frac{10}{\sqrt{\pi}} \, \text{cm} = \frac{10}{\sqrt{\pi}} \times 10^{-2} \, \text{m}.\)

Magnetic flux through \(S_1\):

\(⇒ \Phi_{S_1} = B S_1 \cos 180^\circ \\ ⇒ \Phi_{S_1} = 2 \times 10^{-3} \cdot \pi R^2 \cdot (-1) \\ ⇒ \Phi_{S_1} = -2 \times 10^{-3} \cdot \pi \cdot \frac{100}{\pi} \times 10^{-4} \\ ⇒ \Phi_{S_1} = -20 \, \mu\text{Wb}\)

Since the total entering flux equals the total leaving flux:

\(⇒ \Phi_{S_2} = -\Phi_{S_1} = +20 \, \mu\text{Wb}\)

∴ \(\Phi_{S_1} = -20 \, \mu\text{Wb}\) and \(\Phi_{S_2} = +20 \, \mu\text{Wb}.\)

The correct option is 1).

Explanation:

The magnetic field lines are coming in and coming out. There is a zero total flux.

\(\phi_{total}=\phi_{cone}+\phi_{base} =0 \)

The flux through the base is 

\(\phi_{base}=\int B\cdot dA =B_o \pi R^2\)

Then flux through the cone will be :

\(\phi_{cone}=-\phi_{base}=-B_o\pi R^2\)

Thus The correct option is (4): 

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

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

Unit of both Magnetic field strength and Magnetic field intensity is AT/m.

Additional Information

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

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

or, μ0μr = μ0(1 + K)

∴ μr = (1 + K)

Important Points

Absolute Permeability(μ): Absolute permeability is related to the permeability of free space and is a constant value which is given as:

• μ0 = 4π × 10-7 H/m
• Its dimension is [M L T-2 A-2]
• The absolute permeability for other materials can be expressed relative to the permeability of free space as:

           μ = μ0μr

Where μr is the relative permeability which is a dimensionless quantity.

Relative Permeability (μr): Relative permeability for a magnetic material is defined as the ratio of absolute permeability to absolute permeability of air. It is a unitless quantity.

Susceptibility (K): It is the ratio of the intensity of magnetization (I) to the magnetic field strength (H). It is a unitless quantity.

B-H curve:

Definition:- The B-H curve is the characteristic representation of the magnetic properties of a material when placed in an external magnetic field.

For non-magnetic materials, B-H curve will not saturate. The curve will have a fixed slope approximately equal to µ₀. Therefore it will have a straight line passing through origin.

Note:-

• Diamagnetic materials have a slightly smaller slope.
• Paramagnetic materials have a slightly greater slope.

Important Points

• The residual magnetism or remanence or retentivity is the flux density that is left within the material after it has been magnetized.
• The coercivity is the magnetic field intensity that is required to demagnetize the material.
• The saturation effect of the material occurs when all of the magnetic domains within the material have aligned with the external magnetic field.
• This curve gives hysteresis losses in a transformer.

Magnetic flux density:

Magnetic flux density is the amount of flux passing through a defined area that is perpendicular to the direction of the flux, i.e.

Mathematically, this is defined as:

\(B=\frac{ϕ}{A}\)

B = Magnetic flux density

ϕ = Magnetic Flux (weber, Wb)

A = Area (m2)

• SI unit of the magnetic flux density (B) is Tesla (T).
• Tesla is equivalent to one weber per meter squared or Weber/meter2 (Wb / m2)
• The CGS unit of B is gauss where 1 gauss = 10-4 tesla.

Important Notes:

• Tesla is the SI unit of magnetic flux density.
• Magnetic field strength is one of two ways that the intensity of a magnetic field can be expressed.
• Technically, a distinction is made between magnetic field strength H, measured in amperes per meter (A/m), and magnetic flux density B, measured in Newton-meters per ampere (Nm/A), also called Tesla (T).
• 1 Tesla equals to 10⁴ Gauss. The smaller unit being gauss. 

Concept

The magnetizing force (H) for a circular conductor is given by:

\(H={I\over 2π d}\)

where, I = Current & d = Distance

The flux density (B) is given by:

B = μ_oμ_r × H

μ_o = Absolute Permeability

μ_r = Relative Permeability

Calculation

Given,  I = 100π A

d = 10 cm

\(H={100π\over 2π 10× 10^{-2}}\)

H = 500 AT/m

For air, μr = 1

B = 4π × 10^-7 × 500 

B = 6.28 × 10-4 AT/m

Concept:

Let there be a flux in the air gap, then the uniform flux density in the air gap is then given by

\(B = \frac{\emptyset }{A}\)

Since the air gap has been increased by a volume A.dx, the increase in the stored magnetic field energy is,

\(d{W_f} = \frac{{{B^2}}}{{2{\mu _0}}} \times A.dx\)

If the system is ideal with no losses (the motion has taken place slowly from one point of rest to another), the change in magnetic energy must be due to the input of mechanical energy (work done)

\(d{W_M} = d\;{W_f}\)

\(F.dx = \frac{{{B^2}A}}{{2{\mu _0}}}dx\)

\(F = \frac{{{B^2}A}}{{2{\mu _0}}}\) N/m²

Calculation:

Let the cross-sectional area of the iron face be A. Consider the field energy in the airgap volume contained between two parallel faces separated by a distance x.

\({W_f}\left( {B,x} \right) = \frac{1}{2}\frac{{{B^2}Ax}}{{{\mu _0}}}\)
The mechanical force due to the field is

\({F_f} = \frac{{\partial {W_f}\left( {B,x} \right)}}{{\partial x}}\)

\({F_f} = - \frac{1}{2}\frac{{{B^2}A}}{{{\mu _0}}}\)

The negative sign indicates that the force acts in a direction to reduce x (i.e. it is an attractive force between the two faces).

The force per unit area is,

\(\left| {{F_f}} \right| = \frac{1}{2}\frac{{{B^2}}}{{{\mu _0}}}\)

\(\left| {{F_f}} \right| = \frac{1}{2}\frac{{{{\left( {1.5} \right)}^2}}}{{4\pi \times {{10}^{ - 7}}}}\)

\(\left| {{F_f}} \right| = 0.89 \times {10^6}N/m^2\)

Magnetic field strength or field intensity (H) is the amount of magnetising force.

Magnetic flux density (B) is the amount of magnetic force induced on the given body due to the magnetising force H.

The relation between B and H is,

B = μH

Where, μ is the relative permeability

Permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. It is the degree of magnetization that a material obtains in response to an applied magnetic field.

It is a constant of proportionality that exists between magnetic flux density and magnetic field intensity.

Magnetic field strength or field intensity (H) is the amount of magnetising force.

Magnetic flux density (B) is the amount of magnetic force induced on the given body due to the magnetising force H.

The relation between B and H is,

B = μH

Where, μ is the relative permeability

Permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. It is the degree of magnetization that a material obtains in response to an applied magnetic field.

It is a constant of proportionality that exists between magnetic flux density and magnetic field intensity.