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Step-by-Step Solution
Step 1: Recall the Meaning of Magnetic Susceptibility
Magnetic susceptibility ( \chi ) of a material indicates how easily it can be magnetized in response to an external magnetic field. For a paramagnetic material like aluminium, its relative permeability is given by
\mu_r = 1 + \chi.
Step 2: Express the New Magnetic Field Inside the Toroid
When the inside of the toroid is filled with aluminium, the magnetic field B becomes
B = \mu_0 \mu_r H,
where H is the magnetizing field and \mu_0 is the permeability of free space. If B_0 = \mu_0 H is the original magnetic field (when the core was empty or just air), then
B = B_0\,(1 + \chi).
Step 3: Compute the Percentage Increase
The increase in magnetic field from B_0 to B is
\Delta B = B - B_0 = B_0 (1 + \chi) - B_0 = B_0 \, \chi.
Hence, the fractional (relative) increase in the magnetic field is
\frac{\Delta B}{B_0} = \frac{B - B_0}{B_0} = \chi.
To get this as a percentage, multiply by 100:
\frac{\Delta B}{B_0} \times 100 = 100 \,\chi.
Step 4: Match to the Form Given in the Question
We are given \chi = 2.2 \times 10^{-5} for aluminium. Therefore,
\frac{\Delta B}{B_0} = 2.2 \times 10^{-5}, \quad
\text{and as a percentage,} \quad
2.2 \times 10^{-5} \times 100 = 2.2 \times 10^{-3}.
The problem states that the percentage increase in B is written as
\frac{x}{10^4}.
We need to match
2.2 \times 10^{-3} = \frac{x}{10^4}.
Solving for x :
2.2 \times 10^{-3} = \frac{x}{10^4}
\;\Longrightarrow\;
x = 2.2 \times 10^{-3} \times 10^4 = 2.2 \times 10^{1} = 22.
Step 5: Conclude the Value of x
Hence, the value of x (as asked in the question) is
\boxed{22}.
