Unit 2: Electromagnetism

Biot-Savart's Law and Its Applications

Biot-Savart's Law is a fundamental principle in electromagnetism that describes the magnetic field generated by a constant electric current. It serves as the magnetostatics equivalent of Coulomb's Law in electrostatics.

Biot-Savart's Law:
The magnetic field induction dB produced by a small current-carrying element dl carrying current I at a point P located at a distance r is directly proportional to the current, the length of the element, the sine of the angle between the element and the position vector, and inversely proportional to the square of the distance.

Mathematically, the scalar form of the law is written as:

dB = (μ₀ / 4π) × (I dl sin θ / r²)

In vector form, the law is represented as:

dB = (μ₀ / 4π) × (I (dl × r̂) / r²) = (μ₀ / 4π) × (I (dl × r) / r³)

Where:

• μ₀ is the permeability of free space (vacuum), equal to 4π × 10^-7 T·m/A.
• dl is the vector length of the infinitesimal current element.
• r̂ is the unit vector pointing from the current element to the observation point P.
• θ is the angle between the current element vector dl and the displacement vector r.

1. Application to a Straight Conductor

To determine the magnetic field B at a perpendicular distance d from a straight wire carrying a steady current I:

For a wire of finite length bounded by angles φ₁ and φ₂ from the perpendicular line to the wire ends, the integrated magnetic field is:

B = (μ₀ I / 4π d) × (sin φ₁ + sin φ₂)

Special Case: Infinitely Long Straight Conductor

For an infinitely long wire, the angles φ₁ and φ₂ both approach π/2 (90°):

B = (μ₀ I / 4π d) × (sin 90° + sin 90°)
B = (μ₀ I / 4π d) × (1 + 1)
B = μ₀ I / 2π d

Observation: The magnetic field is inversely proportional to the distance d, and its field lines form concentric circles around the wire, determined by the Right-Hand Rule.

2. Application to a Circular Coil Carrying Current

Consider a circular loop of radius R carrying a steady current I. We calculate the magnetic field B along the central axis of the coil at a distance x from its center.

By symmetry, the components of the magnetic field perpendicular to the axis cancel out, while the axial components add together. Integrating around the loop yields:

B = (μ₀ I R²) / (2 (R² + x²)^3/2)

For a coil with N closely wrapped turns, the magnetic field is scaled by N:

B = (μ₀ N I R²) / (2 (R² + x²)^3/2)

Special Case: At the Center of the Circular Coil (x = 0)

B = (μ₀ N I R²) / (2 R³)
B = μ₀ N I / 2R

3. Application to a Solenoid Carrying Current

A solenoid is a long coil of wire wound tightly into a helical shape. We can determine the magnetic field on the axis of a finite solenoid using the Biot-Savart relation by summing up the contributions of individual circular loops. For a solenoid of length L with N turns carrying current I:

B = (μ₀ n I / 2) × (cos θ₁ - cos θ₂)

Where n = N/L is the number of turns per unit length, and θ₁, θ₂ are the angles subtended by the ends of the solenoid at the observation point. For an infinitely long solenoid, θ₁ → 0 and θ₂ → π, giving:

B = μ₀ n I

Divergence and Curl of the Magnetic Field

The behaviors of magnetic fields are mathematically governed by differential vector relations. These equations are fundamental components of Maxwell's equations.

1. Divergence of the Magnetic Field (∇ · B)

The divergence of any magnetic field vector B is always zero at every point in space:

∇ · B = 0

Physical Meaning & Important Observations:

• No Magnetic Monopoles: Unlike electric fields (where ∇ · E = ρ / ε₀), there are no isolated magnetic charges (monopoles). Magnetic poles always exist as dipoles (North and South).
• Continuous Loops: Every magnetic field line that enters a closed volume must also exit it. Consequently, magnetic field lines are continuous, closed loops.
• Solenoidal Field: Any vector field with zero divergence is classified as a solenoidal field.

2. Curl of the Magnetic Field (∇ × B)

The curl of a static magnetic field describes its rotation or circulation and is directly related to the local electric current density J:

∇ × B = μ₀ J

Physical Meaning:

• Electric currents are the active physical sources of the curl (circulation) of magnetic fields.
• This equation represents the differential (local) form of Ampere's Circuital Law.
• If there is no electric current density in a region (J = 0), the curl of the magnetic field is zero (∇ × B = 0), making the field irrotational in that specific region.

Magnetic Vector Potential

Just as the electrostatic field E can be represented using a scalar potential V (where E = -∇V), the magnetic field B can be represented using a vector field because of its solenoidal nature.

Definition

Since the divergence of the curl of any vector field is mathematically zero (∇ · (∇ × A) = 0), and we know that ∇ · B = 0, we can define a vector field A, called the Magnetic Vector Potential, such that:

B = ∇ × A

The SI unit of the magnetic vector potential is Tesla-meter (T·m) or Weber per meter (Wb/m).

Properties and Gauge Transformations

• Gauge Freedom: The vector potential A is not uniquely defined because we can add the gradient of any scalar function λ to it without altering the magnetic field B (since the curl of any gradient is zero, i.e., ∇ × ∇λ = 0). Thus, A' = A + ∇λ gives the same magnetic field.
• Coulomb Gauge Condition: To make A unique in magnetostatics, we impose the Coulomb gauge constraint:

  ∇ · A = 0

• Vector Poisson's Equation: Substituting B = ∇ × A into Ampere's law (∇ × B = μ₀ J) and using the vector identity ∇ × (∇ × A) = ∇(∇ · A) - ∇²A, we get:

  ∇(∇ · A) - ∇²A = μ₀ J

  Applying the Coulomb gauge (∇ · A = 0) simplifies this to the vector Poisson equation:

  ∇²A = -μ₀ J

Ampere's Circuital Law and Its Applications

Ampere's Circuital Law provides a highly symmetric method to calculate magnetic fields for symmetric current distributions.

Ampere's Circuital Law Statement:
The line integral of the magnetic field B around any closed loop (called an Amperian loop) is equal to μ₀ times the net current I_enc enclosed by that loop.

∮ B · dl = μ₀ I_enc

1. Application to an Infinitely Long Solenoid

Consider an infinitely long, tightly wound solenoid with n turns per unit length carrying a current I.

To find the field inside, construct a rectangular Amperian loop abcd where:

• Side ab of length L lies inside the solenoid, parallel to the axial magnetic field.
• Sides bc and da are perpendicular to the solenoid axis.
• Side cd is located outside the solenoid, where the magnetic field is negligible (B ≈ 0).

Let's evaluate the line integral ∮ B · dl:

∮ B · dl = ∫_a^b B · dl + ∫_b^c B · dl + ∫_c^d B · dl + ∫_d^a B · dl

• For segment ab: B and dl are parallel, so ∫ B · dl = B × L.
• For segments bc and da: B is perpendicular to dl, so ∫ B · dl = 0.
• For segment cd: B ≈ 0 outside, so ∫ B · dl = 0.

The total integral is:

∮ B · dl = B × L

The total enclosed current is the number of turns in length L multiplied by the current per turn:

I_enc = n × L × I

Applying Ampere's Law:

B × L = μ₀ × (n × L × I)
B = μ₀ n I

2. Application to a Toroidal Coil (Toroid)

A toroid is a solenoid bent into a circular, closed ring. Let N be the total number of turns and I be the current carrying through the wire. Due to axial symmetry, the magnetic field lines form concentric circles inside the toroid core.

We choose a circular Amperian loop of radius r concentric with the toroid. The line integral of B is:

∮ B · dl = B ∮ dl = B × (2π r)

Let's evaluate the three separate regions of space:

1. Region 1: In the open space interior to the toroid (r < r_inner)
   The loop encloses no current. Hence, I_enc = 0:

   B = 0

2. Region 2: Inside the core of the toroid (r_inner < r < r_outer)
   The loop cuts through all N turns of the coil. The total enclosed current is N × I. Applying Ampere's Law:

   B × (2π r) = μ₀ N I
   B = μ₀ N I / 2π r

3. Region 3: Outside the toroid (r > r_outer)
   The loop encloses equal and opposite currents (as each turn of wire enters and exits the plane of the loop). The net enclosed current is zero. Hence:

   B = 0

Magnetic Properties of Materials

When materials are placed in an external magnetic field, they respond dynamically due to the alignment of atomic-level magnetic dipoles. To mathematically quantify this response, several key parameters are defined.

1. Magnetic Intensity (H)

Magnetic intensity (also called the magnetizing field strength) represents the external magnetic excitation applied to a medium. It depends only on the free currents and geometry, independent of the material media.

H = B₀ / μ₀

Where B₀ is the magnetic field in a vacuum. The unit of H is Ampere per meter (A/m).

2. Magnetic Induction (B)

Magnetic induction (or magnetic flux density) represents the total macroscopic magnetic field inside the material. It includes both the external field and the field induced by the magnetic dipoles of the material (quantified by Magnetization M):

B = μ₀ (H + M)

Where M is the Intensity of Magnetization (magnetic dipole moment per unit volume). The unit of B is Tesla (T) or Weber per square meter (Wb/m²).

3. Magnetic Permeability (μ)

Permeability measures the degree to which a substance can be penetrated or magnetized by an applied magnetic field.

• Absolute Permeability (μ): The ratio of magnetic induction B to the magnetic intensity H inside the medium:

  μ = B / H

• Relative Permeability (μ_r): A dimensionless ratio of the medium's permeability to the permeability of free space:

  μ_r = μ / μ₀

4. Magnetic Susceptibility (χ_m)

Magnetic susceptibility is a dimensionless quantity that measures how easily a material magnetizes in response to an external magnetizing field. It is the ratio of the induced magnetization M to the magnetic intensity H:

χ_m = M / H

Relation Between μ_r and χ_m

We can establish the mathematical bridge between relative permeability and susceptibility as follows:

B = μ₀ (H + M)

Substitute M = χ_m H:

B = μ₀ (H + χ_m H) = μ₀ H (1 + χ_m)

Since B = μ H and μ = μ₀ μ_r, we write:

μ H = μ₀ H (1 + χ_m)
μ₀ μ_r H = μ₀ H (1 + χ_m)

Dividing both sides by μ₀ H gives the fundamental relation:

μ_r = 1 + χ_m

Classification of Magnetic Materials

Magnetic materials are broadly classified into three categories—Diamagnetic, Paramagnetic, and Ferromagnetic—depending on their behavior when subjected to external fields.

Important Concepts & Common Exam Pitfalls

• The Superconductor Exception: Superconductors are classified as perfect diamagnetic materials because they exhibit complete expulsion of magnetic fields (Meissner Effect) below their critical temperature. Their relative permeability μ_r = 0 and susceptibility χ_m = -1.
• Common Mistake: Confusing B and H. Remember that H represents the external magnetic driving force (created by currents), whereas B represents the actual total magnetic flux density inside the medium, which includes the material's magnetic response (magnetization).
• Curie Temperature (T_c): This is the temperature at which a ferromagnetic material loses its domain alignment and behaves as a standard paramagnetic material.