Magnetostatic Field MCQ Quiz - Objective Question with Answer for Magnetostatic Field - Download Free PDF

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:

The magnetic field outside a toroidal coil is zero. Let's delve into the detailed explanation and reasoning behind this conclusion.

Understanding the Toroidal Coil:

A toroidal coil, or simply a toroid, is a coil of wire in the shape of a doughnut or a closed loop. The wire is wound around a circular core, and the core itself can be either solid or hollow. The purpose of this design is to confine the magnetic field within the core and minimize the magnetic field outside the coil.

Magnetic Field in a Toroidal Coil:

To understand why the magnetic field outside a toroidal coil is zero, we need to consider the principles of electromagnetism, specifically Ampère's Law. Ampère's Law states that the line integral of the magnetic field B around any closed loop is proportional to the total current I passing through the loop. Mathematically, it is expressed as:

B × dl = μ₀ × I

where:

• B is the magnetic field
• dl is a differential element of the loop
• μ₀ is the permeability of free space
• I is the current passing through the loop

In the case of a toroidal coil, the wire is wound in such a way that the current flows in circular loops around the core. Because the coil is wound symmetrically and continuously around the core, the magnetic field inside the core is also circular and confined within the core. The magnetic field lines inside the toroid follow the loops of the wire, creating a strong and uniform magnetic field within the core.

Why the Magnetic Field Outside is Zero:

Outside the toroidal coil, the contributions of the magnetic field from each loop of the wire cancel each other out. This is due to the symmetry and closed-loop nature of the toroid. For any point outside the toroid, there are equal and opposite magnetic field contributions from different sections of the coil, resulting in a net magnetic field of zero.

Mathematically, this can be understood by applying Ampère's Law to a closed loop that lies outside the toroid. Since no net current passes through this loop (the current entering the loop at one point is exactly balanced by the current leaving the loop at another point), the line integral of the magnetic field around this loop is zero. Consequently, the magnetic field outside the toroid must be zero to satisfy Ampère's Law.

Therefore, the correct option is:

1) The magnetic field outside a toroidal coil is zero.

Explanation:

Curl of a Vector Field

Definition: The curl of a vector field is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. For a vector field A, the curl is denoted as ∇ × A. It can be interpreted as the amount of rotation or the 'twisting' of the field at a point.

Mathematical Expression: The curl of a vector field A = (A_x, A_y, A_z) in Cartesian coordinates is given by:

∇ × A = ( (∂A_z/∂y - ∂A_y/∂z), (∂A_x/∂z - ∂A_z/∂x), (∂A_y/∂x - ∂A_x/∂y) )

Properties of the Curl:

• The curl of a vector field is itself a vector field.
• The divergence of the curl of any vector field is zero: ∇·(∇ × A) = 0.
• The curl of a gradient of any scalar field is zero: ∇ × (∇V) = 0.
• The curl of a curl of a vector field A is given by: ∇ × (∇ × A) = ∇(∇·A) - ∇²A.

Correct Option Analysis:

The correct option is:

Option 3: The curl of a scalar field V, (∇ × V), makes sense.

This option is incorrect because the curl of a scalar field does not make sense in the context of vector calculus. The curl operator applies to vector fields, not scalar fields. For a scalar field V, taking the curl ∇ × V is undefined. The concept of curl requires a vector input, and since V is a scalar, ∇ × V does not have a meaningful interpretation.

Additional Information

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

Option 1: The divergence of the curl of a vector field vanishes, that is, ∇·(∇ × A) = 0.

This statement is true. It is a fundamental property of the curl operator that the divergence of the curl of any vector field is always zero. This can be proven using vector identities and is a key concept in vector calculus.

Option 2: The curl of a vector field is another vector field.

This statement is true. When you take the curl of a vector field, the result is another vector field that describes the rotation or circulation of the original field at each point.

Option 4: The curl of the gradient of a scalar field vanishes, that is, ∇ × (∇V) = 0.

This statement is also true. For any scalar field V, the curl of its gradient is always zero. This can be shown mathematically and is another fundamental property in vector calculus.

Conclusion:

Understanding the properties of vector calculus operators such as the gradient, divergence, and curl is essential for solving problems in fields such as electromagnetism and fluid dynamics. The curl operator, in particular, applies only to vector fields and describes the rotation or twisting of the field. The incorrect option in this context is the one that incorrectly implies that the curl of a scalar field makes sense, which it does not.

Solution and Explanation:

We are given the following options to identify the correct vector identity:

• Option 1: ∇^(2) \overrightarrow{\mathrm{ A}} = ∇(∇ ⋅ \overrightarrow{\mathrm{A}}) - ∇ × ∇ × \mathrm{A}
• Option 2: ∇ ×(∇ V) ≠ 0
• Option 3: ∇ ⋅(∇ × A) ≠ 0
• Option 4: ∇ ⋅(\vec{A} + \vec{B}) ≠ ∇ ⋅ \vec{A} + ∇ ⋅ \vec{B}

Option 1 is the correct identity. This is known as the vector identity for the Laplacian of a vector field. Let's break down why it is correct.

The Laplacian of a vector field \overrightarrow{\mathrm{A}} is defined as:

∇^(2) \overrightarrow{\mathrm{A}} = (∇ ⋅ ∇) \overrightarrow{\mathrm{A}}

We can further expand this identity using vector calculus identities:

The Laplacian of a vector field \overrightarrow{\mathrm{A}} can be written as:

∇^(2) \overrightarrow{\mathrm{A}} = ∇(∇ ⋅ \overrightarrow{\mathrm{A}}) - ∇ × ∇ × \overrightarrow{\mathrm{A}}

This identity is derived from the vector calculus identity which states that the Laplacian of a vector field can be decomposed into the gradient of the divergence of the vector field minus the curl of the curl of the vector field.

To understand this better:

• Gradient (∇): Measures the rate and direction of change in a scalar field.
• Divergence (∇ ⋅): Measures the magnitude of a source or sink at a given point in a vector field.
• Curl (∇ ×): Measures the rotation of a vector field.

Thus, the correct identity is option 1 because it represents the correct decomposition of the Laplacian of a vector field into its constituent operations.

Concept

The work done by a force on an object is calculated by taking the dot product of the force vector and the displacement vector. It represents the energy transferred to the object by the force.

$W=q(\overrightarrow{F}.\overrightarrow{d})$

Calculation

Given, q = 1nC

F = 4ax - 3ay + 2a_z

d = 10ax + 2ay - 7a_z

$W=1\times {10}^{-9}[(4a_x-3a_y+2a_z).(10a_x+2a_y-7a_z)]$

W = 40 - 6 -14

W = 20 nJ

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

In an electric magnetic circuit, for establishing a magnetic field, energy must be spent, through no energy is required to maintain it.

Let us take on the example of exciting coils of electromagnetic as shown in the figure:

The energy supplied to it is spent in two ways:

• Part of it goes to meet I²R losses and is lost once for all.
• Part of it goes to create a flux and is solved in the form of the magnetic field as “potential energy” and is similar to the potential energy of a raised weight when a mass ‘M’ is raised through height ‘H”, the potential energy solved it is MgH.
• Work is done in raising the mass, but once the mass is raised to a certain height, no further expenditure of energy is required to maintain it at that position.
• This mechanical potential energy can be recovered to get electrical energy which is stored in the form of a magnetic field.

• Magnetic field strength is also called as the magnetic intensity. 
• The magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts.
• Magnetic field strength arises from an external current and is not intrinsic to the material itself.
• It may be defined in terms of magnetic poles: one centimetre from a unit pole the field strength is one oersted.
• The magnetic field strength is defined by the unit of Oersted (Oe) or Ampere/metre (A/m). When it is defined by flux density, the units of Gauss (G) or Tesla (T) are used.

Concept:

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.

Additional Information

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.

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

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) cause 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)

Susceptibility (K) is normally considerd as unitless quantity.

Note: Sometimes the value of susceptibility is given in terms of H/m and denoted by (K').

And, K' = Kμ

∴ ${\mu }_r=1+\frac{K^{\prime }}{{\mu }_0}$

Magnetic field strength (B):

• Magnetic field strength refers to the ratio of the MMF which is required to create a certain Flux Density within a certain material per unit length of that material. Some experts also call is as the magnetic field intensity.
• Furthermore, magnetic flux refers to the total number of magnetic field lines that penetrate an area. Furthermore, the magnetic flux density tends to diminish with increasing distance from a straight current-carrying wire or a straight line that connects a pair of magnetic poles around which the magnetic field is stable.
• Magnetic field strength refers to a physical quantity that is used as one of the basic measures of the intensity of the magnetic field. The unit of magnetic field strength happens to be ampere per meter or A/m.
• As the distance of a point from the magnet increases the magnetic flux density decreases and the magnetic field intensity decreases. 

​The magnetic intensity due to an isolated pole of strength mp at a distance (r) is:

$B=\frac{{\mu }_o{m}_p}{4\pi r^2}$

Hence option (4) is the correct answer.

The inductance is given by:

$L=\frac{\mu N^{2}A}{l}$

Where,

N = Number of turns

A = Cross-sectional area 

l = length 

μ = Permeability

Energy stored by the inductor is given by: 

$E=\frac{1}{2}L{I}^2$

Where, I = Current flowing through the inductor

Note that, Inductor stores energy in the form of magnetic field.

∴ Energy stored in a magnetic field of length l metre and of cross-section A sq-m is given by:

$E=\frac{1}{2}L{I}^2$

$E=\frac{1}{2}\times \mu A\frac{N^2}{l}\times I^2$ Joule

Concept:

Magnetic flux density for any current-carrying element is given by:

B = $(\frac{{\mu }_o}{4\pi }\times \frac{I}{r})\times \varphi$

where, B = Magnetic flux density

μ_o = absolute permeability

I = current flowing in current-carrying element

r = distance from current-carrying element to required point

ϕ = angle made by current-carrying element with respect to the required point

Explanation:

B_net = B₁ + B₂ + B₃

Magnetic flux densities B₁ and B₃ are equal to zero as the current flowing is equal and opposite in nature.

So, B_net will be due to circular loop only.

Hence, ϕ = 2π

Bnet =  B₂

B_net = $(\frac{{\mu }_o}{4\pi }\times \frac{I}{r})\times 2\pi$

Bnet = $\frac{{\mu }_o}{2}\frac{I}{r}$

Fleming's Right-hand Rule: 

This rule shows the direction of induced current when a conductor attached to a circuit moves in a magnetic field. It can be used to determine the direction of current in a generator's windings.

Fleming’s left-hand Rule:

Whenever a current-carrying conductor is placed in a magnetic field, the conductor experiences a force which is perpendicular to both the magnetic field and the direction of the current.

In an electric motor, the electric current and magnetic field exist (which are the causes), and they lead to the force that creates the motion (which is the effect), and so the left-hand rule is used.

Right-hand thumb rule:

• The Right-hand thumb rule is used to determine the direction of magnetic field around a current carrying wire.
• This rule states that "when an electric current passes through a straight wire that is held by the right hand with the thumb pointing upwards and the fingers curling up the wire, the thumb points in the direction of the conventional current (from positive to negative), and the fingers point in the direction of the magnetic field".
• Below figure represents the direction of current and magnetic field through a straight current carrying conductor.

Concept:

The force between two long parallel conductors is given by:

$\frac{F}{l}=\frac{{\mu }_0}{2\pi }\frac{I_1{I}_2}{d}$

Where F is the force

l is the length

I1 and I2 are the currents flowing through the two conductors

d is the distance between the conductors

The force between two long parallel conductors is inversely proportional to the distance between the conductors.

The nature of force depends on the direction of current, i.e.

1) If the current flowing in both the wires is in the same direction, the two wires will attract each other.

2) If the current flowing in both the wires is in the opposite direction, the two wires will repel each other.

Calculation:

Given: I₁ = I₂ = 4 A, d = 6 mm = 6 × 10^-3 m, l = 0.6 m

The mutual force between the conductors will be:

$F=\frac{{\mu }_0}{2\pi }\frac{l\times I_1{I}_2}{d}$

$F=\frac{{\mu }_0}{2\pi }\frac{0.6\times 4\times 4}{6\times {10}^{-3}}$

$F=\frac{4\pi \times {10}^{-7}}{2\pi }\frac{0.6\times 4\times 4}{6\times {10}^{-3}}$

Solving the above, we get:

F = 3.2 × 10-4 N

Since the current flowing in both the wires is in the opposite direction, the two will repel each other.

Notes:

Case 1: Two conductors are in parallel and this arrangement is connected in series with dc supply.

During this arrangement total current I from supply flows in two paths I1 for conductor 1) and I2 for conductor 2). Thus current in both the wire flow in some direction thus there would be an attractive force between these two conductors.

Case 2: Two conductors are in series & this arrangement is connected in series with dc supply.

During this arrangement, the total current I flows through the two conductors. Since they are connected in series connection, the same current flows in both conductors but the direction of the current will be in opposite direction. Thus there would exist repulsive force between these two conductors.

The correct answer is Soft Iron.

• Soft iron is selected for making the core of an electromagnet because the core is used in a solenoid for the production of the strongest magnetism.

Additional Information

•  The strongest continuous manmade magnetic field, 45 T, was produced by a hybrid device, consisting of a Bitter magnet inside a superconducting magnet.
  • The resistive magnet produces 33.5 T and the superconducting coil produces the remaining 11.5 T.
• The magnetic field is strongest at the centre and weakest between the two poles just outside the bar magnet.
  • The magnetic field lines are densest at the centre and least dense between the two poles just outside the bar magnet.
  • The intensity of the field is weakest near the equator where it is horizontal.
  • The magnetic field's intensity is measured in gauss.
  • The magnetic field has decreased in strength through recent years.