A paramagnetic substance in the form of a cube with sides 2 cm from a magnetic dipole moment of `30xx10^(-6)J//T` when a magnetic intensity of `80xx10^(3)A//m` is applied. Its magnetic susceptibility is found to be `Nxx10^(-4)`. After calculate the value of N is found to be ........

Explanation

To solve the problem, we need to calculate the magnetic susceptibility of the given paramagnetic substance using the provided information. Here’s a step-by-step solution:

### Step 1: Understand the Given Data
- Side of the cube (s) = 2 cm = 0.02 m
- Magnetic dipole moment (μm) = 30 x 10^(-6) J/T
- Magnetic intensity (H) = 80 x 10^(3) A/m

### Step 2: Calculate the Volume of the Cube
The volume (V) of the cube can be calculated using the formula:
\[ V = s^3 \]
Substituting the value of s:
\[ V = (0.02 \, \text{m})^3 = 0.000008 \, \text{m}^3 = 8 \times 10^{-6} \, \text{m}^3 \]

### Step 3: Calculate the Intensity of Magnetization (I)
The intensity of magnetization (I) is given by:
\[ I = \frac{\mu_m}{V} \]
Substituting the values:
\[ I = \frac{30 \times 10^{-6} \, \text{J/T}}{8 \times 10^{-6} \, \text{m}^3} = \frac{30}{8} \times 10^{0} = 3.75 \, \text{A/m} \]

### Step 4: Calculate the Magnetic Susceptibility (χ)
The magnetic susceptibility (χ) is defined as:
\[ \chi = \frac{I}{H} \]
Substituting the values:
\[ \chi = \frac{3.75 \, \text{A/m}}{80 \times 10^{3} \, \text{A/m}} = \frac{3.75}{80000} = 0.000046875 \]

### Step 5: Express the Susceptibility in the Required Form
The problem states that the magnetic susceptibility is of the form \( N \times 10^{-4} \). To express \( 0.000046875 \) in this form:
\[ 0.000046875 = 4.6875 \times 10^{-5} \]
To convert this to the required form:
\[ 4.6875 \times 10^{-5} = 0.46875 \times 10^{-4} \]
Thus, \( N = 0.46875 \).

### Final Answer
The value of \( N \) is approximately \( 0.47 \).

---