Electromagnetic Induction Test - 1

CONCEPT:

Eddy Current:

• When a changing magnetic flux is applied to a bulk piece of conducting material then circulating currents is called eddy currents are induced in the material.
• Because the resistance of the bulk conductor is usually low, eddy currents often have large magnitudes and heat the conductor.

EXPLANATION:

• Speedometers: In a speedometer, a magnet rotates with the speed at the vehicle.
• The magnet is placed inside an aluminum drum which is carefully pivoted and held in position by a hairspring.
• As the magnet rotates, eddy currents are set up in the drum which opposes the motion of the magnet.
• A torque is exerted on the drum in the opposite direction which deflects the drum through an angle depending on the speed of the vehicle

​Mutual induction:

• When an electric current is passed through a coil changes with time, an emf is induced in the nearby coil then this phenomenon is called mutual induction.
• The transformers work on mutual induction.

Induced emf in the second coil due to primary coil is given by,

Where,

M = coefficient of mutual inductance

I1 = current in the primary coil in A.

t = time in second (s).

Conclusion:

Mutual inductance is the phenomenon by which an induced e.m.f. is produced in a coil due to change in the electric current or the magnetic flux associated with a different, neighboring coil.

Electromotive force (emf):

• It is an equal terminal potential difference when no current flows.
• Faraday first law: Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced.
• Faraday's Second Law: States that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil.  i.e

CONCEPT:

• According to Faraday's Law, induced emf in a coil is given by:

Induced emf \(=-N \frac{Δ ϕ }{Δ t}\)

• Lenz's law: It states that the direction of the induced current in the coil will be always in such a way that it opposes the change which produces the current.
  • Lenz's Law is just a small addition to Faraday's law.
  • Look at the negative sign in the formula.
  • A negative sign shows the opponent.

EXPLANATION:

• Lenz's law: It states that the direction of the induced current in the coil will be always in such a way that it opposes the change which produces the current.
• So the correct answer is option 3.

CONCEPT:​

• Faraday's Law: Any change in the magnetic flux of wire will cause a voltage (induced emf) to be "induced" in the coil.
  • This change can be produced by changing the magnetic field strength, moving the coil into or out of the magnetic field,  moving a magnet toward or away from the coil, rotating the coil relative to the magnet, etc.

Induced emf \(=-N \frac{Δ ϕ }{Δ t}\)

CALCULATION:

Let the number of turns in a coil is 1.

induced emf  \(=- \frac{Δ ϕ }{Δ t}\)

The magnitude of induced emf \(= \frac{Δ ϕ }{Δ t}= \frac{3t^2 + 2t}{\Delta t}=6t+2\)

at t = 1 sec

induced emf = (6t+2)_{at t= 1} = 6 × 1 + 2 = 8

So, the correct answer is Option 1, i.e 8 V

• Lenz's law says that the direction of the induced current in the coil will be always in such a way as to oppose the change which produces the current.
• It is just a small addition to Faraday's law. See negative signs in the formula. A negative sign shows the opponent.

CONCEPT:

Faraday's first law of electromagnetic induction

• Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced.
• If the conductor circuit is closed, a current is induced which is called induced current.

​Faraday's second law of electromagnetic induction

• The induced emf in a coil is equal to the rate of change of flux linked with the coil.

\(\Rightarrow e=-\frac{d\text{ }\!\!\Phi\!\!\text{ }}{dt}\)

Where dΦ = change in magnetic flux and e = induced e.m.f.

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

​EXPLANATION:

• The magnetic flux associated with a coil is given as,

\(⇒ \phi=BAcosθ\)     -----(1)

where B = magnetic field, A = area and θ = 0°

• When a coil of copper wire is parallel to a uniform magnetic field, the magnetic flux associated with the coil is given as,

⇒ θ = 90°

\(⇒ \phi=BAcos90\)

\(⇒ \phi=0\)

• Since the coil is moving parallel to a uniform magnetic field so the magnetic flux associated with the coil will remain zero during the whole time.
• And the magnetic flux associated with the coil is not changing during the motion, so there will be no emf induced in the coil. Hence induced emf in the coil will be zero.
• Hence, option 3 is correct.

CONCEPT:

Lenz's Law:

• According to this law, the emf will be induced in a coil due to a changing magnetic flux in such a way that it will oppose the cause which has produced it.
• This law states that the induced emf in a conductor due to a changing magnetic flux is such that the magnetic field created by the induced emf opposes the change in a magnetic field.

\(\Rightarrow emf=-N\left ( \frac{d\phi}{dt} \right )\)

where N = number of loops and dϕ =  Change in magnetic flux

• The above equation is given by Faraday's law, but the negative sign is a result of Lenz's law.

EXPLANATION:

• If the magnet is given some initial velocity towards the coil and is released, it will slow down because the emf will be induced in a coil due to a changing magnetic flux in such a way that it will oppose the cause which has produced it. Here the induced emf will try to stop the magnet.
• It can be explained as the following:
• The current induced in the coil will produce heat.
• From the energy conservation, if heat is produced, there must be an equal decrease in some other form of energy. Here is the kinetic energy of the moving magnet.
• So we can say that Lenz's law is associated with the conservation of energy.
• Hence, option 4 is correct.

Given that, magnetic flux density (B) = 30 T

Area (A) = 200 cm²

Emf (E) = 100 V

Number of turns (N) = 100

Change in magnetic flux \(d\phi = BA\cos \theta = 30 \times 200 \times {10^{ - 4}} \times \cos 0 = 0.6\)

We know that, \(E = N\frac{{d\phi }}{{dt}}\)

\(\begin{array}{l} \Rightarrow 100 = 100 \times \frac{{0.6}}{{dt}}\\ \Rightarrow dt = 0.6{\rm{\;}}sec \end{array}\)

CONCEPT:

• AC Generator: An electrical machine used to convert mechanical energy into electrical energy is known as ac generator/alternator.

EXPLANATION:

• AC Generator: It works on the principle of electromagnetic induction i.e., when a coil is rotated in a uniform magnetic field, an induced emf is produced in it. Hence option 1 is correct.

• When the armature coil ABCD rotates in the magnetic field provided by the strong field magnet, it cuts the magnetic lines of force.
• Thus the magnetic flux linked with the coil changes and hence induced emf is set up in the coil.
• The direction of the induced emf or the current in the coil is determined by Fleming’s right-hand rule.

CONCEPT:

• Induced Current: If a conducting loop is exposed to a changing magnetic field, A current can be induced in it. This current is known as induced current.
  • This change in the magnetic field may be produced in several ways; you can change the strength of the magnetic field, move the conductor in and out of the field, alter the length of the distance between a magnet and the conductor, or by changing the area of a loop located in a stable magnetic field.
  • No matter how this change is achieved, the result, an induced current, is always the same.
  • The strength of the current will vary in proportion to the change of magnetic flux.

EXPLANATION:

• The induced emf and induced current are generated due to change in the magnetic field.
• Due to the change in the magnetic field, the magnetic flux is generated.
• This flux leads to the generation of induced emf according to Faraday's law.
• And due to induced emf, an induced current is generated.
• So the correct answer is option 2.

Concept:

• Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force. It consists of two laws.
  • Faraday's First law of Electromagnetic Induction: Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced. If the conductor circuit is closed, a current is induced which is called induced current.
  • Faraday's Second law of Electromagnetic Induction: The induced emf in a coil is equal to the rate of change of flux linkage.
• Joule's law of heating: Amount of heat produced by a steady electric current through a conductor is proportional to resistance of the conductor, to the square of current and to the duration of current.  

            H = K.R.I².t  where, R = resistance of conductor, I = constant current flowing through a conductor, t = duration of current flow.

• Coulomb's law: It states that the force between two point charges Q1 and Q₂

            1) acts along the line joining the two point charges

            2) is directly proportional to product (Q1Q2) of the two charges

            3) is inversly proportional to the square of the distance between them

                 F ∝ \(\frac{Q_{1}Q_{2}}{R^2}\) where, F = force between charges, Q₁,Q₂ = two point charges, R = distance between them.

• Ampere's law: It states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.

            \(\mathop \sum \nolimits {\bf{B\Delta l}} = {\bf{μ I}}\) where, B = magnetic field, l = length of element, μ = permeability, I = electric current