Faraday's Law MCQ Quiz - Objective Question with Answer for Faraday's Law - Download Free PDF

Concept:

In a magnetic circuit, magnetic flux (\( \phi \)) behaves similarly to electric current in an electrical circuit. In a series magnetic circuit, all the magnetic components (like cores and air gaps) are arranged in series, and the same flux flows through each part, regardless of the varying reluctance of the individual sections.

Key Principle:

Just as current remains the same in a series electrical circuit, magnetic flux remains constant in a series magnetic circuit.

Evaluation of Options:

Option 1: the same – Correct
Same flux flows through all elements in a series magnetic circuit.

Option 2: different –  Incorrect
Flux would differ only in parallel magnetic paths, not in series.

Option 3: zero – Incorrect
Flux exists as long as there is magnetomotive force (mmf).

Option 4: infinite –  Incorrect
Infinite flux is not physically possible.

Concept:

According to Lenz’s Law, the direction of the induced current is always such that it opposes the cause producing it. In a transformer, when current flows in the secondary winding, it generates a magnetic field that opposes the magnetic field of the primary coil.

This opposition is what maintains energy conservation and proper transformer action. The effect of this opposing magnetic field is referred to as a demagnetizing effect.

Concept

The average EMF induced in the coil is given by:

\(E=-N{Δ ϕ \over Δ t}\)

The magnetic flux is given by:

\(ϕ = {NI\over R}\)

where, E = EMF

N = No. of turns

Δϕ = Change in flux

Δt = Change in time

I = Current

R = Reluctance

Calculation

Given, N = 100

I = 1 A

R = 1000 AT/Wb

When the current reverses, the flux changes from $+0.1$ Wb to $-0.1$ Wb.

The change in flux is given by:

Δϕ = ϕ_final - ϕ_initial

Δϕ = (−0.1) − (0.1) = −0.2 Wb

\(E=-(100)× {-0.2 \over 10× 10^{-3}}\)

E = 100 × 20 × 10^-3 V

E = 2 V

Explanation:

Motional EMF in a Conductor

Definition: Motional EMF (Electromotive Force) is the voltage generated across a conductor when it moves through a magnetic field. This phenomenon is a direct consequence of Faraday's Law of Electromagnetic Induction, which states that a change in magnetic flux through a circuit induces an EMF in the circuit.

Working Principle: The principle behind motional EMF can be understood through the Lorentz force. When a conductor moves through a magnetic field, the free charge carriers (such as electrons) within the conductor experience a force due to the magnetic field. This force causes the charge carriers to accumulate on one end of the conductor, creating a potential difference (voltage) across the conductor. The magnitude of the motional EMF (ε) is given by the equation:

ε = B × L × v × sin(θ)

where:

• B is the magnetic field strength.
• L is the length of the conductor within the magnetic field.
• v is the velocity of the conductor.
• θ is the angle between the velocity of the conductor and the magnetic field.

The correct option is: The resistance of the conductor

Explanation:

In the quasi-static approximation, the magnetic field inside the toroid is given by:

\( B = \frac{\mu_0 N I}{2 \pi s}, \quad \text{(inside the toroid)},\)

and outside the toroid, the magnetic field is zero:

\(B = 0, \quad \text{(outside the toroid)}\).

The magnetic flux Φ  through the loop inside the toroid is the integral of the magnetic field B over the path of the loop:

\(Φ = \int B \, dl = \frac{\mu_0 N I}{2 \pi} \int_{a}^{a+w} \frac{ds}{s}.\)

Evaluating the integral:

\(Φ = \frac{\mu_0 N I h}{2 \pi} \ln\left( 1 + \frac{w}{a} \right).\)

For small w/a , we can approximate the logarithmic term:

\(\ln\left( 1 + \frac{w}{a} \right) \approx \frac{w}{a}.\)

Thus, the magnetic flux becomes:

\( Φ \approx \frac{\mu_0 N I h w}{2 \pi a}.\)

Step 2: Find the Rate of Change of Flux

The current I is increasing at a constant rate \(\frac{dI}{dt} = k\) , so the rate of change of the magnetic flux is:

\(\frac{dΦ}{dt} = \frac{\mu_0 N h w}{2 \pi a} \cdot k = \frac{\mu_0 N h w k}{2 \pi a}.\)

Step 3: Induced Electric Field

From Faraday's law, the induced electric field E is related to the rate of change of magnetic flux:

\(E = -\frac{\mu_0 N h w k}{2 \pi a} \cdot \frac{a^2}{(a^2 + z^2)^{3/2}},\)

which simplifies to:

\( E = -\frac{\mu_0 N h w k a}{4 \pi (a^2 + z^2)^{3/2}}.\)

Note:

This induced electric field is analogous to the magnetic field produced by a circular current, which is given by:

\(B = \frac{\mu_0 I}{2 (a^2 + z^2)^{3/2}} \hat{z},\)

where I is the current and a and z are the radius and distance, respectively, from the center of the loop.

Thus, the induced electric field has a similar form to the magnetic field of a circular current and is given by:

\( E = -\frac{\mu_0 N h w k a}{4 \pi (a^2 + z^2)^{3/2}}.\)

This electric field behaves similarly to the magnetic field of a current loop, with the same \((a^2 + z^2)^{-3/2}\) dependence.

The correct answer is 1): \( E = -\frac{\mu_0 N h w k a}{4 \pi (a^2 + z^2)^{3/2}}.\)

Dynamically induced EMF: When the conductor is rotating and the field is stationary, then the emf induced in the conductor is called dynamically induced EMF.

Ex: DC Generator, AC generator

Static induced EMF: When the conductor is stationary and the field is changing (varying) then the emf induced in the conductor is called static induced EMF.

Faraday's laws: Faraday performed many experiments and gave some laws about electromagnetism.

Faraday's First Law:

Whenever a conductor is placed in a varying magnetic field an EMF gets induced across the conductor (called induced emf), and if the conductor is a closed circuit then induced current flows through it.
A magnetic field can be varied by various methods:

• By moving magnet
• By moving the coil
• By rotating the coil relative to a magnetic field

Faraday's second law of electromagnetic induction states that the magnitude of induced emf is equal to the rate of change of flux linkages with the coil.

According to Faraday's law of electromagnetic induction, the rate of change of flux linkages is equal to the induced emf:

\({\rm{E\;}} = {\rm{\;N\;}}\left( {\frac{{{\rm{d\Phi }}}}{{{\rm{dt}}}}} \right){\rm{Volts}}\)

Faraday's first law of electromagnetic induction:

It 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:

It states that the magnitude of the voltage 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.​

\(v=-N\frac{d\text{ }\!\!\Phi\!\!\text{ }}{dt}\)

Where N = number of turns, dΦ = change in magnetic flux and v = induced voltage.

The negative sign says that it opposes the change in magnetic flux which is explained by Lenz law.

The correct answer is option 3): 1250 V 

Concept:

The Inductance of the coil is given by

L = \(N \phi \over I\) Henry

EMF . induced E = L\(di \over dt\) V

Calculation:

L = \(1000 ×0.25 × 10^{-3}\over 2\)

= 0.125

E = 0.125× \((10 -0) \over 1 \times 10 ^{-3}\)

(Where current changes from 10A to 0 A)

= 1250 V

Explanation:

• Faraday’s first law states that whenever there is a change in the magnetic flux linked with a coil or a conductor, an electromotive force (EMF) is induced in the coil. This law describes the fundamental principle of electromagnetic induction, stating that the magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux.
• In the given statement, it describes the generation of EMF when a conductor or coil moves in a magnetic field, causing a change in the magnetic flux linked with the conductor.
• This change in flux induces an EMF in the conductor, in accordance with Faraday’s first law.
• Hence, the statement aligns with Faraday’s first law of electromagnetic induction

CONCEPT:

Lenz's Law:

• According to this law, the direction of induced emf or current in a circuit is such as to oppose the cause that produces it.
• This law gives the direction of induced emf/induced current.
• This law is based upon the law of conservation of energy.

EXPLANATION:

• Laplace's law indicates that the tension on the wall of a sphere is the product of the pressure times the radius of the chamber and the tension is inversely related to the thickness of the wall. Therefore the option 1 is incorrect.

• According to Lenz's law, the direction of induced emf or current in a circuit is such as to oppose the cause that produces it. Therefore the option 2 is correct.
• Fleming's right-hand rule shows the direction of induced current but it gives no relation between the direction of induced emf or current in a circuit is such as to oppose the cause that produces it. Therefore the option 3 is incorrect.
• This law is also known as loop rule or voltage law (KVL) and according to it “the algebraic sum of the changes in potential in a complete traversal of a mesh (closed-loop) is zero”, i.e. Σ V = 0. Therefore the option 3 is incorrect.

Faraday’s first law:

• 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:

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.

\(E = N\frac{{d\phi }}{{dt}}\)

Important Points

Method to change the magnetic field:

• By moving a magnet towards or away from the coil.
• By moving the coil into or out of the magnetic field.
• By changing the area of a coil placed in the magnetic field.
• By rotating the coil relative to the magnet.

Faraday’s Law states that a change in magnetic flux induces an emf in a coil.

Also, Lenz’s Law states that this induced emf produces a flux which opposes the flux that generates this emf, i.e.

\(emf=-\frac{d\phi }{dt}\)   ---(1)

EMF is also defined as:

\(emf = \mathop \oint \nolimits_c \overset{\rightharpoonup}{E} .~\overset{\rightharpoonup}{{dl}}\)

Also, \(\phi ~\left( {Net~Flux} \right) = \mathop \smallint \nolimits_s \overset{\rightharpoonup}{B} .d\overset{\rightharpoonup}{s} \)

Putting the above in Equation (1), we get:

\(\mathop \oint \nolimits_c \overset{\rightharpoonup}{E} .d\overset{\rightharpoonup}{l} = - \frac{{\partial \phi }}{{\partial t}} = - \frac{\partial }{{dt}}\mathop \smallint \nolimits_s \overset{\rightharpoonup}{B} .d\overset{\rightharpoonup}{s}\)

\( \mathop{\int }_{S}\left( \nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} \right).d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s}=-\int \frac{\partial B}{\partial t}.d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s}\)

\(\nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E}=-\frac{\partial B}{\partial t}\)

Concept:

According to Faraday's law, the induced emf in a coil (having N turns) is the rate of change of magnetic flux linked with coil,

\({\rm{e}} = {\rm{-N}}\frac{{{\rm{d}}ϕ }}{{{\rm{dt}}}}\)

N = number of turns in the coil

ϕ = magnetic flux link with the coil

Calculation:

Given that ϕ = (t² – 3t) m-wb and N = 200

Induced emf in coil

 \({\rm{e}} = {\rm{-N}}\frac{{{\rm{d}}ϕ }}{{{\rm{dt}}}}\)

\({\rm{e}} = -200\frac{{\rm{d}}}{{{\rm{dt}}}}\left( {{{\rm{t}}^2} - 3{\rm{t}}} \right)×10^{-3}\)

e = -200 (2t - 3) × 10^-3 

then the induced emf in the coil at t = 4

e = - 200 (2 × 4 - 3) × 10^-3 = - 1 V

Concept:

• One of the important properties of the dielectric material is its permittivity. Permittivity ϵ is the measure of the ability of a material to be polarized by an electric field.
• ϵ is complex and is frequency dependent. The imaginary part corresponds to a phase shift of polarization relative to “E” and leads to the attenuation of EM waves passing through the medium.
• When a dielectric is subjected to an alternating electric field then, the dielectric loss will be;

\(W\left( t \right)=\frac{1}{2}\omega {{\epsilon }_{0}}{{\epsilon }_{\text"{r{}}}}~E_{0}^{2}~W/{{m}^{3}}\)

Where, ω = Angular frequency

ϵ₀ = Absolute permittivity

ϵ_r“ = Imaginary part of complex relative permittivily

E₀ = Peak Voltage.