Problem 75 - College Physics

Question

\(\cdot\) Platinum is a paramagnetic metal having a relative perme- ability of 1.00026 (a) What is the magnetic permeability of platinum? (b) If a thin rod of platinum is placed in an external magnetic field of 1.3500 \(\mathrm{T}\) , with its axis parallel to that field, what will be the magnetic field inside the rod?

Step-by-Step Solution

Answer

(a) \( \mu \approx 1.25696438 \times 10^{-6} \, \text{T}\cdot\text{m/A}\); (b) \(\mathbf{B_i} \approx 1.350351 \, \text{T}\).

Step 1: Understand Relative Permeability

Relative permeability (\( \mu_r \)) is a dimensionless factor which describes how permeable a material is to magnetic fields compared to vacuum (\( \mu_0 \) = \(4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A} \)). Here, we are given that \( \mu_r = 1.00026 \) for platinum.

Step 2: Calculate Magnetic Permeability

The magnetic permeability (\( \mu \)) of a material can be calculated by multiplying its relative permeability (\( \mu_r \)) by the permeability of free space (\( \mu_0 \)). Use the formula: \( \mu = \mu_r \times \mu_0 \). Thus, for platinum: \[\mu = 1.00026 \times 4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A} \approx 1.00026 \times 1.256637061 \times 10^{-6} \, \text{T}\cdot\text{m/A} \approx 1.25696438 \times 10^{-6} \, \text{T}\cdot\text{m/A}\].

Step 3: Understand Internal Magnetic Field in the Rod

A material's internal magnetic field (\(\mathbf{B_i}\)) inside an external magnetic field (\(\mathbf{B_0}\)) is given by the product of the external field and the relative permeability: \(\mathbf{B_i} = \mu_r \times \mathbf{B_0}\). Here, \(\mathbf{B_0} = 1.3500 \, \text{T}\) and \(\mu_r = 1.00026\).

Step 4: Calculate the Magnetic Field Inside the Platinum Rod

Substitute the given values into the formula: \[\mathbf{B_i} = 1.00026 \times 1.3500 \, \text{T} = 1.350351 \, \text{T}.\]Thus, the magnetic field inside the rod is approximately \(1.350351 \, \text{T}\).

Key Concepts

Relative Permeability

In the realm of magnetism, the relative permeability is a crucial parameter. It's represented by the symbol \(\mu_r\) and describes how a material responds to a magnetic field. Specifically, it indicates how much more or less susceptible a material is to magnetization compared to free space, which has a permeability \(\mu_0\). The permeability of free space is a universal constant \(\mu_0 = 4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}\).
Relative permeability is dimensionless because it is a ratio. Since \(\mu_r = \frac{\mu}{\mu_0}\), if a material has a \(\mu_r\) greater than 1, it is more susceptible to magnetization than a vacuum. Conversely, if \(\mu_r\) is less than 1, the material is less likely to be magnetized. This concept is fundamental in understanding how materials interact with magnetic fields and form the foundation for analyzing magnetic properties in engineering and physics.

• A value of \(\mu_r = 1\) means the material behaves like a vacuum.
• Values greater than 1 indicate that the material can be magnetized (e.g., paramagnetic or ferromagnetic materials).
• Values less than 1 imply diamagnetic behavior, where the material is repelled by magnetic fields.

Paramagnetic Materials

Paramagnetic materials, like platinum in this example, have a slightly higher tendency to become magnetized when exposed to an external magnetic field. These materials are characterized by having a relative permeability slightly greater than 1 but not by much. They possess weak, positive susceptibility to magnetic fields.
Such materials have unpaired electrons, which align with external magnetic fields. This alignment enhances the internal magnetic field but only slightly. As an outcome of this weak interaction, when the external field is removed, these materials do not retain the magnetization. This behavior is due to the thermal motion of atoms that quickly disrupts the alignment of magnetized domains.
Here are some key properties of paramagnetic materials:

• They have a relative permeability just above 1, indicating slight magnetization.
• Their susceptibility to magnetism is positive but weak.
• They do not exhibit permanent magnetization once the external field is removed.
• Common paramagnetic materials include aluminum, oxygen, and the platinum from our exercise.

Magnetic Field Calculations

Calculating the magnetic field inside a material involves understanding how the external magnetic field interacts with the material's properties. For our exercise, where a platinum rod is placed inside a magnetic field, the magnetic field inside the rod can be calculated using the formula: \[\mathbf{B_i} = \mu_r \times \mathbf{B_0}\]Where \(\mathbf{B_i}\) is the internal magnetic field, \(\mu_r\) is the relative permeability, and \(\mathbf{B_0}\) is the external magnetic field.
For the platinum rod, if the external magnetic field \(\mathbf{B_0}\) is 1.3500 \(\text{T}\) and the relative permeability \(\mu_r\) is 1.00026, substituting these values into the formula gives:\[\mathbf{B_i} = 1.00026 \times 1.3500 \ \text{T} = 1.350351 \ \text{T}\]This calculation shows that the internal field \(\mathbf{B_i}\) is very slightly stronger than the external field.
Such calculations are crucial in designing materials and systems in electromagnetics and can help predict how a material will behave in a given magnetic environment.

• Use the formula \(\mathbf{B_i} = \mu_r \times \mathbf{B_0}\) for internal fields.
• Understanding these calculations aids in engineering applications like transformers and magnetic sensors.