Definitions [16]
The total number of magnetic field lines passing perpendicularly through a given surface area.
Electromagnetic induction is the production of an electromotive force across an electrical conductor in a changing magnetic flux or magnetic field.
The property of a coil by which it opposes the change in its own current and induces an emf in itself — numerically equal to the ratio of magnetic flux (produced due to current in the circuit) linked with the circuit to the current flowing in it, or the ratio of induced emf produced around the circuit to the rate of change of current in it — is called self-inductance.
OR
The self-inductance (L) of a coil is defined as the ratio of the total magnetic flux linkage through the coil to the current flowing in it. Equivalently, it equals the magnitude of the induced EMF per unit rate of change of current.
Define self-inductance.
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.
Define the coefficient of self-induction.
It is defined as magnetic flux linked with the solenoid when unit current flows through it.
Define mutual inductance.
The mutual inductance (M) of two circuits (or coils) is the magnetic flux (Φs) linked with the secondary circuit per unit current (IP) of the primary circuit.
The property of two coils by which a change in current in one coil induces an emf in the other coil — equal to the magnetic flux linked with one circuit per unit current in the other, or the value of induced emf produced in the secondary circuit per unit rate of change in current in the primary circuit — is called mutual inductance.
OR
Mutual Inductance (M) of a pair of coils is defined as the ratio of the total magnetic flux linkage in the secondary coil to the current in the primary coil that produces it.
The coefficient of coupling K between two coils is the fraction of the total magnetic flux produced by one coil that links with the other coil.
An electrical device which converts low alternating voltage at high current to high alternating voltage at low current (or vice versa) — i.e., a device which reduces or increases the voltage in an AC circuit through mutual induction — is called a transformer.
Define a Transformer.
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.
An a.c. generator is a device which converts the mechanical energy into the electrical energy using the principle of electromagnetic induction.
When a charged capacitor is allowed to discharge through a non-resistive inductor, electrical oscillations of constant amplitude and frequency are produced; these are called L-C oscillations.
The current flowing in a purely inductive or purely capacitive circuit for which cos φ = 0 and no power is dissipated even though the current is flowing is called wattless current.
OR
Wattless Current (also called reactive current) is the component of current in a purely inductive or purely capacitive circuit that flows without consuming any power.
In the expression Pav = VrmsIrms cos ϕ, the quantity cos φ is called the power factor.
OR
Power Factor is the cosine of the phase angle (φ) between the voltage and current in an AC circuit. It equals the ratio of true (average) power to apparent power.
Give any one definition of power factor.
The Power Factor is the ratio of True Power (measured in Watts) to Apparent Power (measured in Volt-Amperes) in an AC circuit.
Power factor (cos Φ) = `"True power"/"Apparent power"`
The direction of the induced EMF (and hence the induced current) in a closed conducting loop is always such that it opposes the change in magnetic flux that produced it.
Formulae [13]
ΦB = \[\vec B\] ⋅ \[\vec A\] = B A cos θ
| Symbol | Meaning | SI Unit |
|---|---|---|
| \[Φ_B\] | Magnetic Flux | Weber (Wb) |
| B | Magnetic Field Strength | Tesla (T) |
| A | Area of the surface | m² |
| θ | Angle between B and the normal to the surface | degrees/radians |
L = \[\frac{N\Phi_B}{I}\]
ε = -L\[\frac{dI}{dt}\]
Where:
| Symbol | Meaning | SI Unit |
|---|---|---|
| L | Self-inductance (coefficient) | Henry (H) |
| N | Number of turns in the coil | — |
| \[Φ_B\] | Magnetic flux through one turn | Weber (Wb) |
| I | Current through the coil | Ampere (A) |
| ε | Induced EMF (self-induced) | Volt (V) |
| dI/dt | Rate of change of current | A s⁻¹ |
M = K\[\sqrt {L_1L_2}\]
Where:
- L1, L2 = Self-inductances of coil 1 and coil 2
- K = Coefficient of coupling (dimensionless, no units)
- Range: 0 ≤ K ≤ 1
Therefore:
M ≤ \[\sqrt {L_1L_2}\]
N2ϕ21 ∝ I1 ⟹ N2ϕ21 = M ⋅ I1
Therefore:
\[\frac{V_s}{V_p}=\frac{N_s}{N_p}=\frac{I_p}{I_s}=k\]
Pinput = Poutput
\[\frac{I_p}{I_s}=\frac{N_s}{N_p}=\frac{V_s}{V_p}\]
\[\frac{V_s}{V_p}=\frac{N_s}{N_p}=k\]
Where:
- Vs = Secondary (output) voltage (V)
- Vp = Primary (input) voltage (V)
- Ns = Number of turns in secondary coil
- Np = Number of turns in primary coil
- k = Transformation ratio (turns ratio)
\[Z=\sqrt{R^2+(X_L-X_C)^2}\]
P = VI = \[\frac{V_mI_m}{2}[\cos\phi-\cos(2\omega t+\phi)]\]
Where:
- Vm = Peak voltage
- Im = Peak current
- ϕ = Phase angle between voltage and current
- ω = Angular frequency
- t = Time
\[\cos\phi=\frac{P_{av}}{P_{apparent}}=\frac{P_{av}}{V_{rms}\cdot I_{rms}}\]
Pav = VI cos ϕ = I2Z cos ϕ = \[\frac {V_mI_m}{2}\]cos ϕ = VrmsIrms cos ϕ
The mathematical form of Faraday's law with Lenz's law incorporated is
\[\varepsilon=-N\frac{d\Phi_B}{dt}\]
Theorems and Laws [5]
Faraday's First Law
Whenever the magnetic flux linked with a circuit changes, an EMF is induced in the circuit. The induced EMF lasts only as long as the change in flux is taking place.
Faraday's Second Law
The magnitude of the induced EMF in a circuit is directly proportional to the rate of change of magnetic flux through the surface enclosed by that circuit.
Two circular loops, one of small radius r and the other of larger radius R, such that R >> r, are placed coaxially with centres coinciding. Obtain the mutual inductance of the arrangement.
Let a current IP flow through the circular loop of radius R. The magnetic induction at the centre of the loop is
BP = `(mu_0I_P)/(2R)`
As, r << R, the magnetic induction BP 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, ΦS = MIP
∴ M = `Phi_S/I_P = (mu_0pir^2)/(2R)`
Statement: The mutual inductance of coil 1 with respect to coil 2 equals the mutual inductance of coil 2 with respect to coil 1.
This is called the Reciprocity Theorem of Mutual Inductance.
Implication: It does not matter which coil drives the current — the mutual inductance M between the pair is always the same property of the system, not just one coil.youtube
A transformer is based on the principle of mutual induction, i.e., whenever the magnetic flux linked with a coil changes, an emf is induced in the neighbouring coil. For an ideal transformer there is no loss of power, so Pinput = Poutput. On the basis of winding, transformers are of two types — step-up and step-down.
Statement
The induced EMF in a closed loop has a direction such that the current it drives would create a magnetic flux to oppose the change in flux through the circuit.
Mathematically, this is captured by the negative sign:
Proof (Lenz's Law as Conservation of Energy)
Claim: Lenz's law is a necessary consequence of the Law of Conservation of Energy.
Proof by contradiction:
Suppose, contrary to Lenz's law, the induced current aided the change in flux instead of opposing it.
- When the N-pole of a magnet approaches a coil, the induced current (if aiding) would create a South pole on the near face of the coil
- This South pole would attract the incoming North pole of the magnet
- The magnet would accelerate towards the coil without any external effort
- The accelerating magnet would induce more current, which would attract the magnet even more strongly
- This would result in continuously increasing kinetic energy and electrical energy, generated from nothing
- This is a perpetual motion machine — a direct violation of the Law of Conservation of Energy
Since this is impossible, the induced current must oppose the flux change — Lenz's law is proved.
Conclusion
- Lenz's law is not an arbitrary rule — it is mandated by energy conservation
- The work done by the external agent (to overcome the opposing electromagnetic force) is the source of all electrical energy generated
- Without Lenz's law, electromagnetic induction would violate the most fundamental law of physics
Key Points
- Electromagnetic induction requires a changing magnetic flux — a static field produces no induction
- The faster the change in flux, the greater the induced EMF (Faraday's Second Law)
- The induced EMF exists only during the change; it ceases when the flux becomes constant
- Both the motion of a conductor in a magnetic field and the change of current in a nearby circuit can cause induction
- The direction of the induced current can be found using Fleming's Right-Hand Rule or Lenz's Law
Inductive Reactance (XL)
- XL= ωL = 2πfL
- Increases with frequency
- Voltage leads current
Capacitive Reactance (XC)
- \[X_C=\frac{1}{\omega C}=\frac{1}{2\pi fC}\]
- Decreases with frequency
- Current leads voltage
Important Questions [50]
- If ‘R’ is the Radius of Dees and ‘B’ Be the Magnetic Field of Induction in Which Positive Charges (Q) of Mass (M) Escape from the Cyclotron, Then Its Maximum Speed (Vmax) is
- Prove Theoretically Electromagnetic Induction `E = (Dphi)/(Dt)`
- What is electromagnetic induction?
- The Magnetic Induction at a Point Inside the Solenoid Along the Axis
- The Magnetic Flux Through a Loop is Varying According to a Relation Phi = 6t^2 + 7t + 1.Where `Phi` Is In Milliweber and T is in Second. What is the E.M.F. Induced in the Loop at T = 2 Second?
- A Metal Rod Rotates About One of Its Ends Perpendicular to a Plane Whose Magnetic Induction is 4 X 10-3 T.
- The magnetic flux through a loop varies according to the relation Φ = 8t2 + 6t + C, where ‘C’ is constant
- Electric field intensity in free space at a distance ‘r’ outside the charged conducting sphere of radius ‘R’ in terms of surface charge density ‘ a ’ is
- Define the coefficient of self-induction.
- Define self-inductance.
- Explain why the inductance of two coils connected in parallel is less than the inductance of either coil.
- Write the SI Unit and Dimention Of Of Co-efficient of Self Induction
- Explain the Phenomenon of Self Induction.
- The current in a coil changes from 50A to 10A in 0.1 second. The self inductance of the coil is 20H. The induced e.m.f. in the coil is ______.
- The co-efficient of mutual induction between primary and secondary coil is 2H. Calculate induced e.m.f.
- If the Radius of a Sphere is Doubled Without Changing the Charge on It, Then Electric Flux Originating from the Sphere is
- Explain Self Induction and Mutual Induction
- Define Coefficient of Mutual Induction.
- Explain the Phenomenon of Mutual Induction.
- Define mutual inductance.
- An emf of 91 mV is induced in the windings of a coil when the current in o nearby coil is increasing at the rate of 1.3 A/s. what is the mutual inductance (M) of the two coils in mH?
- What is a transformer?
- A circular coil of 100 turns with a cross-sectional area of 1 m2 is kept with its plane perpendicular to the magnetic field of 1 T. The magnetic flux linkage is ______.
- Derive an Expression for Ratio of E.M.F.S and Currents in Terms of Number of Turns in Primary and Secondary Coil.
- Describe the construction and working of a transformer with a neat labelled diagram.
- What is Transformer?
- Distinguish between Step up and Step Down Transformer.
- State the Principle On Which Transformer Works.
- Derive an Expression for E.M.F. and Current in Terms of Turns Ratio
- A Transformer Converts 240 V Ac to 60 V Ac. the Secondary Has 75 Turns. the Number of Turns in Primary Are
- Derive the equation for a transformer.
- Explain the construction and working of the transformer.
- A Coil of 100 Turns, Each of Area 0.02m^2 is Kept in a Uniform Field of Induction 3.5 X10^-5 T.
- Obtain an Expression for the Induced E.M.F. in a Coil Rotating with Uniform Angular Velocity in Uniform Magnetic Field. Plot a Graph of Variation of Induced E.M.F. Against Phase(θ = ωT) Over One Cycle
- Show Graphically the Variation of E.M.F. with Time (T).
- At Which Position of the Plane of the Rotating Coil with the Direction of Magnetic Field, the E.M.F. Induced in the Coil is Maximum ?
- A Capacitor of Capacitance 0.5 μF is Connected to a Source of Alternating E.M.F. of Frequency 100 Hz. What is the Capacitive Reactance? (π = 3.142)
- What is the Inductance of the Coil
- Three Capacitors of Capacities 8 μF, 8 μF and 4 μF Are Connected in a Series and Potential Difference of 120 Volt is Maintained Across the Combination. Calculate the Charge on Capacitor Of Capacity 4 μF.
- Derive an expression for the impedance of an LCR circuit connected to an AC power supply. Draw phasor diagram.
- Discuss the Composition of Two S.H.M.S Along the Same Path Having Same Period. Find the Resultant Amplitude and Intial Phase.
- Six capacitors of capacities 5, 5. 5, 5, 10 and X μ F are connected as shown in the network in the diagram
- If A.C. Voltage is Applied to a Pure Capacitor, Then Voltage Across the Capacitor
- A Solenoid 3.142m Long and 5.0 Cm in Diameter Has Two Layers of Windings of 500 Turns Each and Carries a Current of 5a. Calculate the Magnetic Induction at Its Centre Along the Axis.
- Calculate Total Normal Electric Induction Over the Closed Surface.
- In Series Lcr Circuit at Resonance, Phase Difference Between Current and E.M.F. of Source is
- An A.C. Circuit Consists of Inductor of Inductance 125 Mh Connected in Parallel with a Capacitor of Capacity 50 μF. Determine the Resonant Frequency.
- Obtain an expression for average power dissipated in a purely resistive A.C. circult.
- What is the average value of alternating current over a complete cycle?
- In a series LCR circuit, the phase difference between the voltage and the current is 45°. Then the power factor will be ______.
Concepts [13]
- Introduction to Electromagnetic Induction
- Self Inductance
- Mutual Inductance
- Transformers
- Need for Displacement Current
- Coil Rotating in Uniform Magnetic Induction
- Alternating-Current Generator
- Reactance and Impedance
- LC Oscillations
- Inductance and Capacitance
- Resonant Circuits
- Power in AC Circuit
- Lenz’s Law and Conservation of Energy
