The Steady Magnetic Field

Why it matters

The study of steady magnetic fields is essential in electrical engineering as it forms the basis for understanding how magnetic forces work in devices like transformers, inductors, and motors. These fields are crucial for the design and analysis of electrical machines and systems that are integral to power generation and distribution.

Key ideas

• Steady Magnetic Field: A magnetic field that does not change with time. It is produced by steady currents or permanent magnets.
• Magnetic Flux (Φ): The total magnetic field passing through a given area. It is measured in Weber (Wb).
• Magnetic Flux Density (B): The amount of magnetic flux passing through a unit area perpendicular to the direction of the magnetic field. It is measured in Tesla (T).
• Magnetic Field Intensity (H): The measure of magnetizing force. It is related to the current and the geometry of the current-carrying conductor and is measured in A/m (Ampere per meter).
• Permeability (μ): A measure of how easily a material can support the formation of a magnetic field within itself. It is measured in Henry per meter (H/m).
• Ampere's Law: Relates the integrated magnetic field around a closed loop to the electric current passing through the loop.

Magnetic flux is generally Φ = integral B·dA; Φ = BA assumes uniform B normal to the surface. The scalar relation B = μH assumes a linear isotropic medium; ferromagnets may have nonlinear and history-dependent response. The stated Ampère law is magnetostatic and uses enclosed free current. Transformer action requires time-varying flux, beyond magnetostatics.

Formulas

• Φ = B·A
  • Φ: Magnetic flux (Wb)
  • B: Magnetic flux density (T)
  • A: Area (m²)
• B = μ·H
  • B: Magnetic flux density (T)
  • μ: Permeability (H/m)
  • H: Magnetic field intensity (A/m)
• ∮H·dl = I
  • ∮H·dl: Line integral of magnetic field intensity around a closed path
  • I: Current enclosed by the path (A)

Worked example

Given: A long straight wire carries a current of 10 A. Calculate the magnetic field intensity at a point 0.1 m away from the wire.

1. Use Ampere's Law: ∮H·dl = I
2. For a circular path around the wire: H·(2πr) = I
3. Solve for H: H = I / (2πr)
4. Substitute values: H = 10 A / (2π × 0.1 m)
5. Calculate: H = 15.92 A/m

Final Answer: 15.92 A/m, tangential around the wire with direction from the right-hand rule. The wire is treated as effectively infinite at this distance.

Common mistakes

• Confusing magnetic flux (Φ) with magnetic flux density (B).
• Forgetting to use the correct units, especially when converting between Tesla and Weber.
• Misapplying Ampere's Law by not considering the correct path for the line integral.

For GATE EE

Questions often involve calculating magnetic field intensity using Ampere's Law, determining magnetic flux in a given area, and understanding the relationship between B, H, and μ. Practice problems involving different geometries and current distributions to strengthen understanding.

Quick check

1. What is the unit of magnetic flux density?
2. How is magnetic field intensity related to current in a wire?
3. What does Ampere's Law relate?

Answers: 1. Tesla (T), 2. It is directly proportional to the current, 3. It relates the magnetic field around a closed loop to the current passing through it.