Diamagnetism — Question 10

Question 10

A susceptibility measurement needs a holder-background subtraction

A weak diamagnetic sample and its holder are measured in the same uniform H with negligible demagnetization. The total moment is $m_{\mathrm{tot}}=aH+b$; the empty holder gives $m_h=cH+b$. The sample volume is V and a is less than c.

1. Find the sample susceptibility and explain the role of the intercept b.

2. If the volume estimate is 5 percent too large, find the ratio of inferred to true susceptibility and the relative error in its magnitude.

Question 10 - Solution

A susceptibility measurement needs a holder-background subtraction

(a) Method and calculation

$$m_s=m_{\mathrm{tot}}-m_h=(a-c)H=\chi VH.$$$$\boxed{\chi=(a-c)/V<0};\qquad b-b=0.$$

(b) Interpretation and further calculation

$$V_{\mathrm{est}}=1.05V\Rightarrow\boxed{\chi_{\mathrm{est}}/\chi_{\mathrm{true}}=1/1.05=20/21}.$$$$\boxed{\frac{|\chi_{\mathrm{est}}|-|\chi_{\mathrm{true}}|}{|\chi_{\mathrm{true}}|}=-1/21\simeq-4.762\%}.$$

Physical check

The shared offset cancels, but the holder slope must also be subtracted; otherwise even the inferred sign can be wrong.