Overview of Electromagnetic Induction

• 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).

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.

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.

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.

• 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.

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

or

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

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

e = Blv

• 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).

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

• 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.

• 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.

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 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.

• 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.

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

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

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 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.

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

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

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.

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