Understanding B-H Curves for Magnetic Materials

The question asks about the B-H curve for a material that is a straight line passing through the origin. The B-H curve represents the relationship between the magnetic flux density (B) in a material and the applied magnetic field intensity (H). This curve is a fundamental characteristic that helps us understand the magnetic properties of different materials.

Let's analyze the B-H curves for the given options:

1. Hardened steel: Hardened steel is a ferromagnetic material. Ferromagnetic materials exhibit a phenomenon called hysteresis. Their B-H curves are typically non-linear and form a loop (a hysteresis loop) when the magnetic field is cycled. They have significant remanence (magnetic flux density remaining after the field is removed) and coercivity (field needed to reduce B to zero). Thus, the B-H curve for hardened steel is not a straight line passing through the origin.
2. Silicon steel: Silicon steel is also a ferromagnetic material, often used in transformers due to its high permeability and low hysteresis loss. Like hardened steel, its B-H curve is non-linear and shows hysteresis, although typically with a narrower loop compared to hard magnetic materials like hardened steel. It does not have a straight line B-H curve passing through the origin.
3. Air: Air is a non-magnetic medium, similar to a vacuum. In non-magnetic materials, the relationship between magnetic flux density (B) and magnetic field intensity (H) is linear and is given by the equation:

   \(B = \mu_0 H\)

   Here, \(\mu_0\) is the permeability of free space (or air), which is a constant (\(4\pi \times 10^{-7}\) T·m/A). This equation represents a straight line when plotted on a B-H graph. The line passes through the origin because when \(H=0\), \(B\) is also \(0\). The slope of this line is \(\mu_0\).

4. Soft iron: Soft iron is another ferromagnetic material, known for its high permeability and low coercivity. It is used in applications where the magnetic field needs to be easily established and removed, such as in electromagnets. Its B-H curve is non-linear and exhibits a hysteresis loop, although typically much narrower than that of hardened steel. It does not have a straight line B-H curve passing through the origin.

Based on this analysis, the only material among the options whose B-H curve is a straight line passing through the origin is air, as it behaves essentially like a vacuum in terms of magnetic properties.

Comparing B-H Curves: Magnetic vs. Non-Magnetic Materials

Let's summarize the key differences in B-H curves:

Conclusion on the B-H Curve Shape

The B-H curve for ferromagnetic materials like hardened steel, silicon steel, and soft iron shows a complex relationship between B and H, including hysteresis. This is due to the alignment of magnetic domains within these materials. For non-magnetic materials like air, the magnetic field inside the material is directly proportional to the applied field, resulting in a simple linear relationship \(B = \mu_0 H\). This linear relationship graphically appears as a straight line that starts from and passes through the origin (0,0) on a B-H plot.

Therefore, the material among the given options whose B-H curve is a straight line passing through the origin is air.

Revision Table: B-H Curves Summary

Additional Information on Magnetic Properties and B-H Curves

• Magnetic Permeability (\(\mu\)): It measures a material's ability to support the formation of a magnetic field within itself. For non-magnetic materials like air, it's approximately equal to the permeability of free space, \(\mu_0\). For ferromagnetic materials, permeability is much higher and is not constant; it varies with the magnetic field.
• Relative Permeability (\(\mu_r\)): Defined as \(\mu_r = \mu / \mu_0\). For vacuum/air, \(\mu_r = 1\). For ferromagnetic materials, \(\mu_r \gg 1\). The slope of the B-H curve is related to the permeability. A straight line through the origin for air signifies a constant permeability \(\mu_0\).
• Hysteresis Loop: Represents the lag of magnetic flux density behind the applied magnetic field in ferromagnetic materials. The area of the loop is proportional to the energy lost per unit volume per cycle due to hysteresis.
• Remanence (\(B_r\)): The magnetic flux density that remains in a ferromagnetic material when the applied magnetic field \(H\) is reduced to zero.
• Coercivity (\(H_c\)): The external magnetic field intensity required to reduce the residual magnetic flux density (\(B_r\)) to zero.

Understanding B-H curves is crucial for selecting appropriate magnetic materials for various applications, such as electromagnets, transformers, and permanent magnets.