Unit 3: Electromagnetic Induction and Maxwell's Equations

Faraday's Laws of Electromagnetic Induction

Electromagnetic induction is the physical process of generating an electromotive force (emf) and an induced electric current in a closed loop by changing the magnetic flux linked with that loop over time.

Magnetic Flux (Φ_B) is defined as the total number of magnetic field lines passing normally through a given surface area. Mathematically:
Φ_B = B · A · cos(θ)
where B is the magnetic field strength, A is the surface area, and θ is the angle between the magnetic field vector and the normal (perpendicular) vector to the surface area.

Faraday's First Law

Whenever there is a change in the magnetic flux linked with a closed circuit, an electromotive force (emf) is induced in the circuit. This induced emf exists only as long as the change in magnetic flux is actively occurring.

Faraday's Second Law

The magnitude of the induced electromotive force (emf) in a circuit is directly proportional to the time rate of change of the magnetic flux linked with that circuit.

e = -dΦ_B / dt

For a coil consisting of N closely wound turns, the individual emfs add up in series. The total induced emf is given by:

e = -N (dΦ_B / dt)

The negative sign represents the direction of the induced emf, which is determined explicitly by Lenz's Law.

Important Observation: Emf can be induced in three distinct ways: by changing the magnetic field strength (B), by changing the area of the loop (A) within the magnetic field, or by changing the orientation (θ) of the loop relative to the field lines.

Lenz's Law and Conservation of Energy

Lenz's Law defines the polarity and direction of the induced electromotive force and resulting electric current.

Lenz's Law: The direction of the induced current in a circuit is always such that it opposes the change in magnetic flux that produced it.

Lenz's Law and Conservation of Energy

Lenz's Law is not an independent physical postulate; it is a direct consequence of the Law of Conservation of Energy.

• Opposite Scenario (If Lenz's Law were false): Assume a magnet's North pole is pushed toward a conducting loop. If the induced current created a South pole facing the incoming magnet, the magnet would experience an attractive force. This attraction would cause the magnet to accelerate towards the loop without requiring any external force, generating kinetic and electrical energy indefinitely from zero input. This directly violates the First Law of Thermodynamics.
• The True Physical Scenario: As the North pole of the magnet approaches the loop, the induced current flows in a direction that creates a North pole on the near face of the loop. This creates an opposing repulsive force. To move the magnet closer, an external agent must perform physical work against this magnetic repulsion. This mechanical work is converted directly into the electrical energy of the induced current. Thus, energy is conserved.

Common Mistake: Students often assume that the induced current always opposes the magnetic field itself. It actually opposes the change in magnetic flux. If the magnetic flux is decreasing, the induced current's field will act in the same direction as the external field to sustain the falling flux.

Self-Inductance of a Single Coil

Self-induction is the process by which an opposing electromotive force is induced in a coil when the electric current passing through the same coil changes over time.

The total magnetic flux Φ linked with a coil is directly proportional to the instantaneous current I passing through it:

Φ = L · I

where L is the Self-Inductance or coefficient of self-induction. The SI unit of self-inductance is the Henry (H). Using Faraday's Law, the self-induced back emf is expressed as:

e = -L (dI / dt)

Derivation of Self-Inductance for a Single Solenoid

Consider a long air-core solenoid of length l, cross-sectional area A, and total number of turns N. Let the turn density (turns per unit length) be n = N/l.

1. When a current I flows through the solenoid, the magnetic field B inside its core is uniform:

   B = μ₀ · n · I = μ₀ · (N / l) · I

2. The magnetic flux passing through a single turn of the solenoid is:

   Φ_single = B · A = μ₀ · (N / l) · I · A

3. The total magnetic flux linked with the entire solenoid of N turns is:

   Φ_total = N · Φ_single = μ₀ · (N² / l) · A · I

4. Comparing this expression to the definition Φ_total = L · I, we find the self-inductance L of the single coil:

   L = μ₀ · N² · A / l

If the core is filled with a material of relative permeability μ_r, the self-inductance becomes:

L = μ_r · μ₀ · N² · A / l

Mutual Inductance of Two Coils

Mutual induction is the process by which an induced emf is generated in one coil (secondary coil) due to a time-varying current flowing through a adjacent coil (primary coil).

The magnetic flux Φ_s linked with the secondary coil is directly proportional to the current I_p flowing in the primary coil:

Φ_s = M · I_p

where M is the Mutual Inductance of the two-coil system. The induced emf in the secondary coil is written as:

e_s = -M (dI_p / dt)

Derivation of Mutual Inductance for Two Coaxial Solenoids

Consider two long coaxial solenoids: an inner solenoid S₁ of radius r₁ with N₁ turns, and an outer solenoid S₂ of radius r₂ with N₂ turns, both sharing a length l. Let r₁ < r₂.

1. Let a current I₁ flow through the inner solenoid S₁. The magnetic field B₁ generated inside S₁ is:

   B₁ = μ₀ · (N₁ / l) · I₁

2. Since the field B₁ is confined strictly inside S₁, the magnetic flux from S₁ linked with each turn of S₂ is determined by the cross-sectional area of S₁ (A₁ = π · r₁²):

   Φ_{21_single} = B₁ · A₁ = μ₀ · (N₁ / l) · I₁ · π · r₁²

3. The total magnetic flux linkage of S₂ is:

   Φ_{2_total} = N₂ · Φ_{21_single} = (μ₀ · N₁ · N₂ · π · r₁² / l) · I₁

4. Since Φ_{2_total} = M · I₁, we get the mutual inductance M of the two coils:

   M = μ₀ · N₁ · N₂ · A₁ / l

Energy Stored in a Magnetic Field

An inductor stores electrical energy in its magnetic field. When current is established in an inductor, work must be done against the self-induced back emf.

Step-by-Step Derivation

Let I be the instantaneous current in an inductor of self-inductance L. The magnitude of the opposing back emf is:

e = L (dI / dt)

The power P supplied by the external source to maintain this current against the back emf is:

P = dW / dt = e · I = L · I · (dI / dt)

The small amount of work dW done during an infinitesimal time interval dt is:

dW = L · I · dI

Integrating this work from an initial zero current to a final steady-state current I₀:

W = ∫₀^I₀ L · I · dI = (1/2) · L · I₀²

This work is stored entirely as magnetic potential energy U in the field of the inductor:

U = (1/2) · L · I²

Magnetic Energy Density (u_B)

Magnetic energy density is the energy stored per unit volume within the magnetic field of a solenoid. For a solenoid of cross-sectional area A and length l, the volume is V = A · l.

Substituting the self-inductance L = μ₀ · N² · A / l into the energy formula:

U = (1/2) · (μ₀ · N² · A / l) · I²

Since the magnetic field B is B = μ₀ · N · I / l, we can solve for I as I = B · l / (μ₀ · N). Substituting this back:

U = (1/2) · (μ₀ · N² · A / l) · [B · l / (μ₀ · N)]² = B² · A · l / (2 · μ₀)

Dividing the total energy by the volume V = A · l yields the energy density u_B:

u_B = B² / (2 · μ₀)

Transformer Mechanics and Losses

A transformer is a static electromagnetic device that transfers alternating current electrical energy from one circuit to another at the same frequency, typically changing the operating voltage and current levels via mutual induction.

Working Principle

It consists of two electrically isolated coils wound on a common magnetic soft iron core. The input winding is the Primary Coil (N_p turns) and the output winding is the Secondary Coil (N_s turns).

Applying an alternating voltage V_p to the primary winding creates a continuously changing magnetic flux in the core. This flux links with the secondary coil, inducing an alternating emf V_s. The relationship between voltages, currents, and turns is:

V_s / V_p = N_s / N_p = I_p / I_s = k

where k is the transformer turns ratio. In a Step-up transformer, N_s > N_p (voltage increases, current decreases). In a Step-down transformer, N_s < N_p (voltage decreases, current increases).

Different Losses in a Transformer

In practice, transformers are highly efficient but experience energy losses which are categorized below:

Equation of Continuity of Current

The equation of continuity of current is the mathematical statement of the conservation of electric charge in a continuous medium. It states that any net charge flow out of a closed volume must result in an equal decrease of charge within that volume.

Step-by-Step Derivation

Consider a closed surface S bounding a volume V. Let the local charge density within V be ρ and the current density crossing S be J.

1. The total electric current I leaving the volume through the surface S is:

   I = ∮_S J · dA

2. By the law of conservation of charge, this current must equal the rate of decrease of the total charge Q enclosed by V:

   I = -dQ / dt

3. Expressing total charge Q as a volume integral of charge density ρ:

   ∮_S J · dA = -d/dt [ ∫_V ρ · dV ]

4. For a stationary volume V, the total derivative can be brought inside the integral as a partial derivative:

   ∮_S J · dA = -∫_V (∂ρ / ∂t) · dV

5. Using Gauss's Divergence Theorem to convert the closed surface integral to a volume integral:

   ∫_V (∇ · J) · dV = -∫_V (∂ρ / ∂t) · dV

6. Since this relationship holds true for any arbitrary volume V, the integrands must be identical, yielding the Equation of Continuity:

   ∇ · J = -∂ρ / ∂t

For steady-state currents, the charge density is constant over time (∂ρ / ∂t = 0), reducing the equation to ∇ · J = 0.

Displacement Current

Displacement current is an effective current term postulated by James Clerk Maxwell to resolve a mathematical inconsistency in Ampere's circuital law for time-varying fields.

Inconsistency of Ampere's Circuital Law

Ampere's Circuital Law states that ∮_C B · dl = μ₀ · I_c, where I_c is the conduction current passing through a flat surface bounded by the closed loop C.

Consider a parallel-plate capacitor being charged. Let us define a closed loop C around the wire feeding the capacitor:

• If we select a flat surface S₁ spanning the loop C, the wire pierces it. Thus, the conduction current is I_c, and ∮ B · dl = μ₀ · I_c.
• If we select a curved, balloon-shaped surface S₂ that passes in between the capacitor plates, no wire pierces it. Thus, conduction current is zero (I_c = 0), which implies ∮ B · dl = 0.

This is an physical impossibility because the line integral around the same boundary loop C cannot yield two different values simultaneously.

Maxwell's Resolution and Derivation

Maxwell proposed that the changing electric field inside the capacitor plates generates a magnetic field in the same manner as a conduction current. He defined this as the Displacement Current (I_d).

Let A be the plate area and Q be the instantaneous charge. The electric field E between the plates is:

E = Q / (ε₀ · A)

The electric flux Φ_E passing through the space between the plates is:

Φ_E = E · A = Q / ε₀

Solving for the charge Q:

Q = ε₀ · Φ_E

Differentiating with respect to time gives the displacement current I_d:

I_d = dQ / dt = ε₀ · (dΦ_E / dt)

The corresponding displacement current density is J_d = ε₀ · (∂E / ∂t).

The Ampere-Maxwell Law

Adding the displacement current to the original conduction current yields the modified Ampere-Maxwell Law:

∮ B · dl = μ₀ · (I_c + I_d) = μ₀ · [ I_c + ε₀ · (dΦ_E / dt) ]

This formulation completely resolves the inconsistency: outside the plates, I_c exists and I_d = 0; inside the plates, I_c = 0 and I_d carries the exact same equivalent current, ensuring continuous current flow.

Maxwell's Equations

Maxwell's equations are the four fundamental equations that govern all classical electromagnetic phenomena. They describe how electric charges and currents create electric and magnetic fields, and how those fields interact with each other.

Electromagnetic Wave Propagation and Poynting Vector

Electromagnetic (EM) waves are self-sustaining transverse waves that propagate through empty space or dielectric media. The propagation of EM waves is a direct consequence of Maxwell's third and fourth equations: a changing magnetic field induces a changing electric field, which in turn induces a changing magnetic field, allowing the wave to propagate outward from its source at the speed of light:

c = 1 / √(μ₀ · ε₀)

The Poynting Vector (S)

The Poynting vector is a mathematical vector that represents the rate of energy transport (energy flux) per unit area of an electromagnetic wave. It represents both the magnitude of the power density and the direction of energy propagation.

Poynting Vector (S): The cross product of the electric field vector E and the magnetic field intensity vector H. Mathematically:
S = E × H
Using the relation B = μ₀ · H, the vector can also be written as:
S = (1 / μ₀) · (E × B)

• Direction: The direction of S is perpendicular to the plane containing E and B, which aligns directly with the direction of propagation of the electromagnetic wave.
• SI Unit: Watts per square meter (W/m²).

Time-Average Poynting Vector (Wave Intensity)

Since electromagnetic waves oscillate at extremely high frequencies, direct measurement of the instantaneous Poynting vector is difficult. Instead, we measure its time average over one full wave cycle, which corresponds to the wave intensity (I):

<S> = (1 / 2) · E₀ · H₀ = (1 / (2 · μ₀)) · E₀ · B₀

where E₀ and B₀ are the peak amplitudes of the electric and magnetic fields respectively.