Problem Statement
A material has magnetic susceptibility $\chi = 14\times10^{-5}$ in an applied field $B_0 = 0.5\,\text{T}$. Find its magnetization.
Given Information
- $\chi = 14\times10^{-5}$
- $B_0 = 0.5\,\text{T}$
- $\mu_0 = 4\pi\times10^{-7}\,\text{T m A}^{-1}$
Physical Concepts & Formulas
Magnetization $M = \chi H$, where $H = B_0/\mu_0$ in free space. Paramagnetics have small positive $\chi$; diamagnetics have small negative $\chi$; ferromagnetics have $\chi \gg 1$.
- $M = \chi H$
- $H = B_0/\mu_0$ (free space)
Step-by-Step Solution
Step 1 — Find H: $$H = B_0/\mu_0 = 0.5/(4\pi\times10^{-7}) \approx 3.98\times10^5\,\text{A m}^{-1}$$
Step 2 — Magnetization: $$M = \chi H = 14\times10^{-5}\times3.98\times10^5 = 55.7042\,\text{A m}^{-1}$$
Step 3 — Physical type: $\chi > 0 \Rightarrow$ paramagnetic material; magnetization aligns with applied field.
Worked Calculation
$$M = 14\times10^{-5}\times3.98\times10^5 = 55.7042\,\text{A m}^{-1}$$
Answer
$$\boxed{M = 55.7042\,\text{A m}^{-1}}$$
The small magnetization 55.7042 A/m shows this is a weakly magnetic paramagnetic material. Ferromagnets like iron have $\chi \sim 10^5$, giving magnetizations of $10^6$ A/m.
Physical Interpretation
The small magnetization 55.7042 A/m shows this is a weakly magnetic paramagnetic material. Ferromagnets like iron have $\chi \sim 10^5$, giving magnetizations of $10^6$ A/m.
The magnitude of this result is physically reasonable and consistent with the expected order of magnitude for this class of problem. Comparing with standard values from physical tables confirms we are in the correct range.
This problem illustrates a fundamental principle that appears throughout physics: small changes in one parameter can lead to measurable, predictable changes in the observable quantity. Understanding this relationship is key to experimental design.
Note that the result depends on the square (or square root) of the key variable — this nonlinear dependence is characteristic of many physics phenomena and means that doubling the parameter does not simply double the result. Students should always check dimensional consistency and order-of-magnitude before accepting any numerical answer.
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