Revision: Electromagnetic Induction Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

Whenever there is a change in the number of magnetic field lines linked with a conductor, an electromotive force (e.mf) is developed between the ends of the conductor which lasts as long as there is a change in the number of magnetic field lines through the conductor. This phenomenon is called the electromagnetic induction.

and

Faraday's Definition:

Electromagnetic induction is the phenomenon in which an e.m.f is induced in the coil if there is a change in the magnetic flux linked with the coil.

If the current-carrying conductor is held in the right hand such that the thumb points in the direction of the current, then the direction of the curl of the fingers will give the direction of the magnetic field.

Whenever the number of magnetic lines of force (magnetic flux) passing through a coil changes, an electric current is induced in the coil. This current is called the induced current.

The self-inductance of a circuit is the ratio of magnetic flux (produced due to current in the circuit) linked with the circuit to the current flowing in it. 

It is defined as magnetic flux linked with the solenoid when unit current flows through it.

The mutual inductance M of two circuits (or coils) is the magnetic flux (Φ_s) linked with the secondary circuit per unit current (I_P) of the primary circuit.

The transformer is a device used for converting low voltage into high voltage and high voltage into low voltage. It works on the principle of electromagnetic induction.

The production of an induced emf in a circuit due to a change in current in the same circuit is called self-inductance.

OR

The ratio of magnetic flux linked with a circuit to the current flowing in it is called self-inductance.

OR

The induced emf produced per unit rate of change of current in a circuit is called self-inductance.

The product of number of turns and magnetic flux is called flux linkage.

Mutual inductance is defined as the value of the induced emf produced in the secondary circuit per unit rate of change in current in the primary circuit.

Coefficient of coupling ( K ) is the measure of the portion of flux produced by coil 1 that reaches coil 2, is called coefficient of coupling.

A transformer is a device used to change the voltage of alternating current from low value to high value or vice versa.

The induced emf produced in a motor due to its generator action, which opposes the armature current, is called back emf.

A transformer in which N_s < N_p​ and hence output voltage is less than input voltage is called a step-down transformer.

A transformer in which input power equals output power (no energy loss) is called an ideal transformer.

The induced currents that swirl inside a solid conducting plate due to relative motion between the conductor and the magnetic field are called eddy currents.

The emf induced in a conductor due to its motion in a magnetic field is called motional emf.

A transformer in which N_s > N_p and hence output voltage is greater than input voltage is called a step-up transformer.

\[W=\int\mathrm{d}w=\int_0^ILi\mathrm{d}i=\frac{1}{2}Li^2=U_\mathrm{B}\]

\[u_B=\frac{B^2}{2\mu_0}\]

\[e=-\frac{d\phi}{dt}\]

or

\[e=-\frac{d}{dt}\int_S\mathbf{B}(t)\cdot d\mathbf{a}\]

e = Blv

\[\frac{e_s}{e_p}=\frac{N_s}{N_p}\]

Statement:

When the magnetic flux through a circuit is changing, an induced electromotive force (emf) is set up in the circuit whose magnitude is equal to the negative rate of change of magnetic flux. This is also known as Neumann’s Law.

Mathematical Expression:

If ΔΦ_B​ is the change in magnetic flux in a time interval Δt, then the induced emf e is given by:

e = \[-\frac{\Delta\Phi_B}{\Delta t}\]​​

In the limiting case as Δt → 0:

e = \[-\frac{d\Phi_{B}}{dt}\]​​

• If dΦ_B is in weber (Wb) and dtdtdt in seconds (s), then the emf eee will be in volts (V).
• This equation represents an independent experimental law, which cannot be derived from other experimental laws.

For a tightly-wound coil of N turns, the induced emf becomes:

e = \[-N\frac{d\Phi_B}{dt}\] or e = \[-\frac{d(N\Phi_B)}{dt}\]​

Here, NΦ_B is called the ‘number of magnetic flux linkages’ in the coil, and its unit is weber-turns.

Explanation:

Consider a magnet and a coil:

• When the north pole of a magnet is near a coil, a certain number of magnetic flux lines pass through the coil.
• If either the coil or the magnet is moved, the number of magnetic flux lines (i.e., the magnetic flux) through the coil changes.

Cases:

• Magnet moved away from the coil → Decrease in magnetic flux through the coil.
• Magnet brought closer to the coil → Increase in magnetic flux through the coil.

In both cases, an emf is induced in the coil during the motion of the magnet.

• Faster motion → Greater rate of change of flux → Higher induced emf.
• If both the magnet and coil are stationary, or both are moving in the same direction with the same velocity, there is no change in flux → No induced emf.

Special Case:

• If the coil is an open circuit (i.e., infinite resistance), emf is still induced, but no current flows.
• This shows that it is the change in magnetic flux that induces emf, not current.

Conclusion:

Neumann’s Law establishes that a changing magnetic flux through a circuit induces an emf, and the induced emf is proportional to the rate of change of flux, with a negative sign indicating the direction (as per Lenz’s law).

Limitations:

• The law applies to changing magnetic flux; it does not induce emf if the magnetic flux remains constant.
• No emf is induced if the coil and magnet move together at the same velocity or remain stationary.
• In open circuits, emf is induced, but no current is generated.

Statement:

The direction of the induced emf, or the induced current, in any circuit is such as to oppose the cause that produces it. This law is known as Lenz’s Law.

Explanation / Proof:

• When the north pole of a magnet is moved towards the coil, an induced current flows in the coil in such a direction that the near (left) face of the coil behaves like a north pole.
• Due to the repulsion between the like poles, the motion of the magnet towards the coil is opposed.
• When the north pole of the magnet is moved away from the coil, the induced current flows in such a direction that the near face of the coil becomes a south pole.
• The attraction between opposite poles then opposes the motion of the magnet away from the coil.

In both cases, the induced current opposes the magnet's motion, which is the cause of the current. Therefore, work has to be done to move the magnet, and this mechanical work appears as electrical energy in the coil.

Direction of Induced Current (Fleming’s Right-Hand Rule):

• Stretch the right-hand thumb, forefinger, and middle finger so that they are mutually perpendicular.
• The forefinger points in the direction of the magnetic field.
• The thumb points in the direction of motion of the conductor.
• The middle finger then gives the direction of the induced current.

Conclusion:

Lenz’s Law shows that the induced current always acts in such a direction as to oppose the cause that produces it. This ensures that mechanical energy is converted into electrical energy, and no energy is produced without work being done.

Limitations / Note:

• If the induced current were in a direction that did not oppose the motion of the magnet, electrical energy would be obtained continuously without doing any work, which is impossible.
• Hence, Lenz’s Law is consistent with the principle of conservation of energy.

It is stated that the direction of induced e.m.f. is always in such a direction that it opposes the change in magnetic flux.

e = `(d phi)/(dt)`
Consider a rectangular metal coil PQRS. Let ‘L’ be the length of the coil. It is placed in a partly magnetic field ‘B’. The direction of the magnetic field is perpendicular to the paper and into the paper. The ‘x’ part of the coil is in magnetic field at instant t. If the coil is moved towards the right with a velocity v = dx/dt with the help of an external agent, such as a hand. The magnetic flux through the coil is:

Φ = BA = BLx

∴ Φ = BLx     ...(1)

There is relative motion of a current through the coil. Let ‘i’ be current through the coil.

Three forces act on the coil.
F₁ on conductor PL ∴ F₁ = B_i x, vertically upward.
F₂ on conductor MS ∴ F₂ = B_i x, vertically downward.
F₃ on conductor SP ∴ F₃ = B_i L towards left.
F₁ & F₂ are equal and opposite and also on the same lines. They will cancel each other; F₃ is a resultant force. The external agent has to do work against this force.

∴ F₃ = −B_i l    ...(−ve sign indicates that force is opposite to dx.)
If dx is the displacement in time dt, then the work done (d_w) = F₃ dx.

∴ d_w = − B_iL dx

This power is an electrical energy ‘ei’ where ‘e’ is an induced e.m.f.

∴ e_i = `-(B_i ldx)/(dt)`

∴ e = `-(BLdx)/(dt)`

∴ e = −BLv

∴ e = `-d/dt (BLx)`

∴ e = `(-d phi)/(dt)`    ...[from eq (1)]

Lenz’s Law states that the direction of the induced electromotive force (EMF) and the resulting current in a conductor is always such that it opposes the change in magnetic flux that caused it. 

Mathematically, Lenz’s Law is expressed as:

ε = `(-d phi_B)/dt`

Where,

ε = Induced EMF

Φ_B = Magnetic flux

The negative sign indicates opposition to the change in flux.

Let a current I_P flow through the circular loop of radius R. The magnetic induction at the centre of the loop is

B_P = `(mu_0I_P)/(2R)`

As, r << R, the magnetic induction B_P may be considered to be constant over the entire cross-sectional area of the inner loop of radius r. Hence magnetic flux linked with the smaller loop will be

`Φ_S = B_PA_S = (mu_0I_P)/(2R)pir^2`

Also, Φ_S = MI_P

∴ M = `Phi_S/I_P = (mu_0pir^2)/(2R)`

Statement

The magnitude of the induced emf in a circuit is directly proportional to the rate of change of magnetic flux linked with the circuit.

Mathematical Expression

If ϕ is the magnetic flux linked with the coil at time t, then

e ∝ \[\frac{d\phi}{dt}\]

e = K​\[\frac{d\phi}{dt}\]

Where K is the constant of proportionality.

In SI units, K = 1, therefore,

e = ​\[\frac{d\phi}{dt}\]​

When Lenz’s law is included (to account for direction),

e = −​\[\frac{d\phi}{dt}\]​

For a coil of nnn turns:

e = −n​\[\frac{d\phi'}{dt}\]​

This is also known as the Flux Rule.

Explanation

From experiments, it is observed that:

• Induced emf is produced only when magnetic flux changes.
• A faster change in flux produces a larger emf.
• No emf is produced if flux remains constant.

Thus, the induced emf depends directly on the rate of change of magnetic flux.

The negative sign (from Lenz’s law) indicates that the induced emf opposes the change in magnetic flux.

Conclusion

The induced emf in a circuit is equal to the negative rate of change of magnetic flux linked with it. This quantitative relation is known as Faraday’s Second Law of Electromagnetic Induction or the Flux Rule.

Whenever there is a change of magnetic flux in a closed circuit, an induced emf is produced in the circuit. Also, if a conductor cuts the lines of the magnetic field, an e.mf. is induced between its ends.

This law is a qualitative law as it only indicates the characteristics of induced emf.

The direction of induced current in a circuit is such that the magnetic field produced by it opposes the change in magnetic flux that produces it.

OR

Every effect of induction acts in opposition to the cause that produces it.

• A changing current in a coil produces an induced emf that opposes the change in current.
• Energy is stored in the magnetic field of an inductor when current flows through it.
• Inductance depends on the coil’s size, shape, number of turns, and core material.
• A long solenoid has higher inductance if it has more turns per unit length and a larger cross-sectional area.
• In a series connection, inductance increases, while in a parallel connection, it decreases.

• A change in magnetic flux through a coil produces an induced emf and induced current.
• Induced current is produced only when there is relative motion between the magnet and the coil or when the magnetic field changes.
• The direction of induced current reverses when the direction of motion or polarity of magnet is reversed.
• The magnitude of induced current depends on the speed of motion and the number of turns in the coil.
• Changing current in a primary coil induces current in a nearby secondary coil (principle of mutual induction).

• Motional emf is produced when a conductor moves in a magnetic field.
• Moving the sliding bar increases the loop area and changes the magnetic flux.
• Charges in a moving wire experience the Lorentz force, which induces a current.
• In a rotating rod, different parts move at different speeds, generating an emf.
• Induced emf depends on the change in magnetic flux, whether due to motion or a changing magnetic field.

• Induced emf is produced in a stationary coil when magnetic flux through it changes with time.
• The induced emf is maximum when the rate of change of magnetic flux is maximum.
• According to Lenz’s law, the induced emf is negative when flux increases and positive when flux decreases.
• During one half oscillation of the magnet, two emf pulses are produced — one negative and one positive.
• The peak induced emf is directly proportional to angular amplitude and inversely proportional to time period.

• A generator converts mechanical energy into electrical energy using electromagnetic induction.
• When the armature rotates in a magnetic field, conductors cut magnetic field lines and induce an emf.
• The induced emf varies sinusoidally with time, producing alternating current (AC).
• The maximum induced emf depends on the number of turns, the magnetic field strength, the area of the coil and the angular speed.
• A commutator can convert alternating current into pulsating direct current (DC).

• A motor and generator have similar construction and can function as each other.
• Back emf is produced in a motor due to generator action, and it opposes the armature current (Lenz’s law).
• At the start, back emf is zero, so current is large; as speed increases, back emf increases and current decreases.

• A changing current in one coil produces a changing magnetic flux in a nearby coil and induces an emf in it.

• The magnetic flux linked with one coil is proportional to the current in the other:
  Φ₂₁ = MI₁, Φ₁₂ = MI₂​, and M₁₂ = M_21​.

• The induced emf due to mutual induction is
  e = −M\[\frac {dI}{dt}\]​.

• The unit of mutual inductance is henry (H); 1 H = 1 Ω ⋅ s.

• Mutual inductance depends on the coupling between coils and is given by
  M = K​ \[\sqrt{L_{1}L_{2}}\], where K shows how strongly the coils are coupled.

• Lenz’s law states that the induced current always opposes the change in magnetic flux that produces it.
• When a magnet approaches a loop, the loop develops a similar pole to oppose the magnet's motion.
• Induced current exists only when there is a change in magnetic flux, not when flux is constant.
• Lenz’s law is a direct consequence of the law of conservation of energy.
• The negative sign in Faraday’s law e = -\[\frac{d\phi}{dt}\] represents Lenz’s law mathematically.

• When a loop moves in a magnetic field, magnetic flux changes and an induced emf is produced:
  e = −\[\frac{d\Phi}{dt}\]​
• For the moving loop, flux is
  Φ = BLx
  and the induced emf is
  e = BLv
• Induced current in the loop is
  i = \[\frac{BLv}{R}\]​
• A magnetic force opposes the motion of the loop:
  F = iLB
• Mechanical power equals electrical power (heat produced):
  P = Fv = i²R
  → Energy is conserved.