A bar of an iron-silicon alloy having the B-H behavior shown...

A bar of an iron-silicon alloy having the $B-H$ behavior shown in Figure 20.29 is inserted within a coil of wire $0.40 \mathrm{m}$ long and having 50 turns, through which passes a current of $0.1 \mathrm{A}$. (a) What is the $B$ field within this bar? (b) At this magnetic field, (i) What is the permeability? (ii) What is the relative permeability? (iii) What is the susceptibility? (iv) What is the magnetization?

A bar of an iron-silicon alloy having the $B-H$ behavior shown in Figure 20.29 is inserted within a coil of wire $0.40 \mathrm{m}$ long and having 50 turns, through which passes a current of $0.1 \mathrm{A}$.
(a) What is the $B$ field within this bar?
(b) At this magnetic field,
(i) What is the permeability?
(ii) What is the relative permeability?
(iii) What is the susceptibility?
(iv) What is the magnetization?

Key Concepts

Magnetic Susceptibility (?)

Magnetic susceptibility measures the degree to which a material becomes magnetized in response to an applied magnetic field. It is defined as the ratio of the magnetization (M) of the material to the applied magnetic field strength (H). This parameter is directly related to relative permeability by the relation ?r = 1 + ?, and it helps quantify the magnetic responsiveness of the material.

Magnetization (M)

Magnetization refers to the degree of alignment of magnetic dipoles within a material when exposed to an external magnetic field. It quantifies the internal contribution to the overall magnetic flux density and is influenced by both the intrinsic properties of the material and the applied magnetic field. Measurement of magnetization is key in understanding how materials develop internal magnetic fields in response to external influences.

Magnetic Permeability (?)

Magnetic permeability is a property of a material that quantifies how easily it can support the formation of a magnetic field within itself. It is defined as the ratio of magnetic flux density (B) to magnetic field strength (H). A higher permeability indicates that the material can achieve a greater B for a given H, making it an important parameter in magnetic design and material selection.

Relative Permeability (?r)

Relative permeability is the ratio of a material’s permeability to the permeability of free space (?0). This dimensionless quantity offers a comparative measure of a material's ability to conduct magnetic lines of force relative to a vacuum, and it is pivotal in determining the efficiency of a material in magnetic applications.

Magnetic Flux Density (B)

Magnetic flux density represents the total magnetic field within a material, including both the applied field and the contribution from the material's response. It is typically measured in teslas (T) and is directly influenced by both the magnetic field strength and the intrinsic properties of the material, often expressed as B = ?H where ? is the permeability.

Magnetic Field Strength (H)

Magnetic field strength is a measure of the magnetizing force acting within a material. It is defined in terms of the applied current and geometry of the coil (commonly expressed as H = NI/l, where N is the number of turns, I is the current, and l is the length over which the field is distributed). This value is crucial for determining how a material responds to an external magnetic influence.

Ampere's Law

Ampere's Law connects the magnetic field in a closed loop to the current passing through that loop. In the context of magnetic circuits, this law is used to calculate the magnetic field intensity (H) by relating the number of turns in a coil and the current driving those turns, divided by the length of the magnetic path.

B-H Curve

The B-H curve is a graphical representation of the relationship between magnetic flux density (B) and magnetic field strength (H) for a given material. This curve provides insight into the magnetic behavior of materials, including phenomena such as hysteresis, saturation, and non-linearity. It is essential for understanding how materials like iron-silicon alloys respond to applied magnetic fields.

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