The area of hysteresis loop. of a material is equivalent to `250J / m^2`. When `10 kg ` material is magnetized by an alternating field of 50 Hz then energy lost in one hour will be beta joule. Find `(beta)/ ( I000 ))`. (Density of material is `7.5 / cm^(2)` )

Answer

To solve the problem, we need to calculate the energy lost in one hour due to hysteresis in the given material. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Hysteresis Loss Formula
The energy loss due to hysteresis can be calculated using the formula:
\[ \text{Energy Loss} = \text{Area of Hysteresis Loop} \times \text{Number of Cycles} \times \text{Volume} \]

### Step 2: Calculate the Volume of the Material
Given the mass of the material \( m = 10 \, \text{kg} \) and the density \( \rho = 7.5 \, \text{g/cm}^3 \), we need to convert the density into kg/m³ for consistency:
\[ \rho = 7.5 \, \text{g/cm}^3 = 7500 \, \text{kg/m}^3 \]
Now, we can find the volume \( V \) using the formula:
\[ V = \frac{m}{\rho} = \frac{10 \, \text{kg}}{7500 \, \text{kg/m}^3} = \frac{10}{7500} \, \text{m}^3 = \frac{1}{750} \, \text{m}^3 \]

### Step 3: Calculate the Number of Cycles in One Hour
The material is magnetized by an alternating field of frequency \( f = 50 \, \text{Hz} \). The number of cycles in one second is 50, so in one hour (3600 seconds):
\[ \text{Number of Cycles} = 50 \, \text{cycles/second} \times 3600 \, \text{seconds} = 180000 \, \text{cycles} \]

### Step 4: Calculate the Total Energy Loss
Now we can substitute the values into the energy loss formula:
\[ \text{Energy Loss} = \text{Area} \times \text{Number of Cycles} \times \text{Volume} \]
Substituting the known values:
\[ \text{Energy Loss} = 250 \, \text{J/m}^2 \times 180000 \, \text{cycles} \times \frac{1}{750} \, \text{m}^3 \]
Calculating this:
\[ \text{Energy Loss} = 250 \times 180000 \times \frac{1}{750} \]
\[ = 250 \times 240 = 60000 \, \text{J} \]

### Step 5: Find \( \beta \) and Calculate \( \frac{\beta}{1000} \)
Here, \( \beta \) is the energy lost in one hour, which we calculated as 60000 J. Now we need to find \( \frac{\beta}{1000} \):
\[ \frac{\beta}{1000} = \frac{60000}{1000} = 60 \]

### Final Answer
Thus, the final answer is:
\[ \frac{\beta}{1000} = 60 \]