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Qwestrum Engineering360 · Electrical & Electronics · Electromagnetic Theory

Magnetostatic Fields

Magnetostatics finds the field of steady currents; Ampere’s law ∮H·dl = I_enc gives the field of symmetric geometries (long wire, solenoid, toroid), and the Lorentz force gives the force on moving charges and conductors.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • MagneticfluxΦ=∫B⋅dA;nomagneticmonopolesMagnetic flux \Phi = \int B \cdot dA; no magnetic monopoles
  • InductanceL=NΦI;energyW=12LI2Inductance L = N \frac{\Phi}{I}; energy W = \frac{1}{2} L I^{2}
  • μ_r for ferromagnetic materials — non-linear B-H curve

Topic details

Introduction

For symmetric current distributions, Ampere’s circuital law is far faster than Biot–Savart. A long straight wire gives H = I/2πr; an ideal solenoid gives H = nI (turns per metre times current); a toroid gives H = NI/2πr.

Scope in B.Tech and GATE syllabus

The Biot–Savart law is reserved for finite or irregular geometries where symmetry fails, such as a current loop on its axis. The force relations F = qv×B and F = IL×B underpin motor action and force between conductors.

Key relations & formulas

Formulas (Indian textbook notation)

  • Biot−Savart:dB=(μ04π)Idl×r^/r2Biot-Savart: dB = (\frac{\mu_{0}}{4\pi}) I dl \times r̂ / r^{2}

Formulas (Indian textbook notation)

  • Amperelaw:∮H⋅dl=IenclosedAmpere law: ∮ H \cdot dl = I_{enclosed}

Formulas (Indian textbook notation)

  • Lorentzforce:F=q(v×B);F=IL×BLorentz force: F = q(v \times B); F = I L \times B

Notation and sign conventions

Relation 1 —
Biot−Savart:dB=Biot-Savart: dB =

Formulas (Indian textbook notation)

  • Biot−Savart:dB=(μ04π)Idl×r^/r2Biot-Savart: dB = (\frac{\mu_{0}}{4\pi}) I dl \times r̂ / r^{2}
Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
Amperelaw:∮H⋅dl=IenclosedAmpere law: ∮ H \cdot dl = I_{enclosed}

Formulas (Indian textbook notation)

  • Amperelaw:∮H⋅dl=IenclosedAmpere law: ∮ H \cdot dl = I_{enclosed}
Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
Lorentzforce:F=qLorentz force: F = q

Formulas (Indian textbook notation)

  • Lorentzforce:F=q(v×B);F=IL×BLorentz force: F = q(v \times B); F = I L \times B
Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Inductance links flux to current: L = NΦ/I. For a solenoid L = μ₀μ_r N²A/l, so doubling turns quadruples inductance. Stored magnetic energy is W = ½LI², and energy density is B²/2μ.

Governing relations in practice

The force per unit length between two parallel wires carrying I₁ and I₂ separated by d is F/l = μ₀I₁I₂/2πd, attractive for like directions. This defines the ampere.

Design and analysis considerations

Because ∇·B = 0 there are no magnetic monopoles; field lines always close on themselves, unlike electric field lines that begin and end on charges.

Assumptions and validity limits

State assumptions explicitly before using any relation for magnetostatic fields — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Electromagnetic Theory viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Electromagnetic Theory papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to magnetostatic fields.
4. Use equation 1:
Biot−Savart:dB=Biot-Savart: dB =
.
5. Use equation 2:
Amperelaw:∮H⋅dl=IenclosedAmpere law: ∮ H \cdot dl = I_{enclosed}
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Magnetostatic Fields appears in RF, power apparatus, and communications. In Indian electrical curricula this topic is tested because it connects theory to fields, Maxwell equations, and waves.
GATE and semester exams often combine magnetostatic fields with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use magnetostatic fields?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Applying Ampere’s law to geometries without enough symmetry to pull H out of the integral
• Using total turns N instead of turns-per-metre n for the solenoid field
• Forgetting μ_r for a ferromagnetic core when finding inductance
• Dropping the vector cross-product direction in the Lorentz force

Quick revision checklist

Before attempting magnetostatic fields problems, confirm you can:
1.
MagneticfluxΦ=∫B⋅dA;nomagneticmonopolesMagnetic flux \Phi = \int B \cdot dA; no magnetic monopoles

2.
InductanceL=NΦI;energyW=12LI2Inductance L = N \frac{\Phi}{I}; energy W = \frac{1}{2} L I^{2}

3. μ_r for ferromagnetic materials — non-linear B-H curve
Revise the solved examples in Elements of Electromagnetics — Matthew Sadiku and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Field inside a long solenoid

Problem

An air-core solenoid has 2000 turns over a length of 0.5 m and carries 3 A. Find H and B inside.

Solution

Turns per metre n = 2000/0.5 = 4000 turns/m.
H = nI = 4000 × 3 = 12000 A/m.
B = μ₀ H = 4π×10⁻⁷ × 12000.
B = 1.2566×10⁻⁶ × 12000 = 0.0151 T ≈ 15.1 mT.

Conceptual check — Magnetostatic Fields

Problem

In a Electromagnetic Theory semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of magnetostatic fields." What should a complete answer include?

Exams & GATE

Sadiku — field inside solenoid, toroid, force between conductors.

📖 Standard books (India)

  • Elements of Electromagnetics — Matthew Sadiku

    Read: Syllabus unit

    Fields, Maxwell equations, and waves

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