# Chapter 5 Summary

# Concepts

- The magnetic force acting on a charged particle q moving with a velocity u in a region containing a magnetic flux density B is F m=q u×B.

- The total electromagnetic force, known as the Lorentz force, acting on a moving charge in the presence of both electric and magnetic fields is F=q(E+u×B).

- Magnetic forces acting on current loops can generate magnetic torques.

- The magnetic field intensity induced by a current element is defined by the Biot–Savart law.

- Gauss’s law for magnetism states that the net magnetic flux flowing out of any closed surface is zero.

- Ampère’s law states that the line integral of H over a closed contour is equal to the net current crossing the surface bounded by the contour.

- The vector magnetic potential A is related to B by B=∇×A.

- Materials are classified as diamagnetic, paramagnetic, or ferromagnetic, depending on their crystalline structure and the behavior under the influence of an external magnetic field.

- Diamagnetic and paramagnetic materials exhibit a linear behavior between B and H with μ≈μ 0 for both.

- Ferromagnetic materials exhibit a nonlinear hysteretic behavior between B and H, and for some, μ may be as large as 10 5 μ 0.

- At the boundary between two different media, the normal component of B is continuous, and the tangential components of H are related by H 2 t−H 1 t=J s, where J s is the surface current density flowing in a direction orthogonal to H 1 t and H 2 t.

- The inductance of a circuit is defined as the ratio of magnetic flux linking the circuit to the current flowing through it.

- Magnetic energy density is given by W m=1 2 μ H 2.

# Mathematical and Physical Models

Maxwell′s Magnetostatics Equations

Gauss’s Law for Magnetism

∇ . B = 0 ⇔ ∮ S B . d s = 0

Ampère’s Law

∇ × H = J ⇔ ∮ C H . d l = I

Lorentz Force on Charge q

F=q(E+u×B)

Magnetic Force on Wire

F m=I∮C d l×B (N)

Magnetic Torque on Loop

T=m×B (N⋅m)
m=n^N I A (A⋅m 2)

Biot–Savart Law

H=I 4 π∫l d l×R ˆ R 2 (A/m)

Magnetic Field

Infinitely LongWire B=ϕ ˆ μ 0 I 2 π r(Wb/m 2)
Circular Loop H=z ˆ I a 2 2(a 2+z 2)3/2(A/m)
Solenoid B≈z ˆ μ n I=z ˆ μ N I l(Wb/m 2)

Vector Magnetic Potential

B=∇×A (Wb/m 2)

Vector Poisson’s Equation

∇2 A=−μ J

Inductance

L=Λ I=Φ I=1 I∫S B⋅d s (H)

Magnetic Energy Density

w m=1 2 μ H 2 (J/m 3)

# Important Terms

Provide definitions or explain the meaning of the following terms:

- Ampère’s law

- Ampèrian contour

- Biot–Savart law

- current density (volume) J

- diamagnetic

- ferromagnetic

- Gauss’s law for magnetism

- hard and soft ferromagnetic materials

- inductance (self- and mutual)

- Lorentz force F

- magnetic dipole

- magnetic energy W m

- magnetic energy density w m

- magnetic flux Φ

- magnetic flux density B

- magnetic flux linkage Λ

- magnetic force F m

- magnetic hysteresis

- magnetic moment m

- magnetic potential A

- magnetic susceptibility χ m

- magnetization curve

- magnetization vector M

- magnetized domains

- moment arm d

- orbital and spin magnetic moments

- paramagnetic

- solenoid

- surface current density J s

- toroid

- toroidal coil

- torque T

- vector Poisson’s equation