To find the ratio of the magnetic induction fields \( B_1, B_2, \) and \( B_3 \) at distances \( r_1 = 30 \, \text{cm}, r_2 = 60 \, \text{cm}, \) and \( r_3 = 90 \, \text{cm} \) from the center of a short magnet, we can follow these steps:
### Step 1: Convert distances from centimeters to meters
We need to convert the distances from centimeters to meters:
- \( r_1 = 30 \, \text{cm} = 0.3 \, \text{m} \)
- \( r_2 = 60 \, \text{cm} = 0.6 \, \text{m} \)
- \( r_3 = 90 \, \text{cm} = 0.9 \, \text{m} \)
### Step 2: Write the formula for magnetic induction
The magnetic induction \( B \) at a distance \( r \) from a short magnet is given by the formula:
\[
B = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^3}
\]
where \( m \) is the magnetic moment of the magnet and \( \mu_0 \) is the permeability of free space.
### Step 3: Establish the ratios for \( B_1, B_2, \) and \( B_3 \)
Since \( m \) and \( \mu_0 \) are constants for all three distances, we can express the ratios as:
\[
\frac{B_1}{B_2} = \frac{r_2^3}{r_1^3}, \quad \frac{B_2}{B_3} = \frac{r_3^3}{r_2^3}, \quad \frac{B_1}{B_3} = \frac{r_3^3}{r_1^3}
\]
Thus, we can write:
\[
\frac{B_1}{B_2} : \frac{B_2}{B_3} : \frac{B_1}{B_3} = \frac{1}{r_1^3} : \frac{1}{r_2^3} : \frac{1}{r_3^3}
\]
### Step 4: Calculate \( r_1^3, r_2^3, \) and \( r_3^3 \)
Now we calculate the cubes:
- \( r_1^3 = (0.3)^3 = 0.027 \, \text{m}^3 \)
- \( r_2^3 = (0.6)^3 = 0.216 \, \text{m}^3 \)
- \( r_3^3 = (0.9)^3 = 0.729 \, \text{m}^3 \)
### Step 5: Write the ratios
Now we can express the ratios:
\[
\frac{B_1}{B_2} : \frac{B_2}{B_3} : \frac{B_1}{B_3} = \frac{1}{0.027} : \frac{1}{0.216} : \frac{1}{0.729}
\]
### Step 6: Simplify the ratios
To simplify, we can take the common denominator (which is \( 729 \)):
- For \( B_1 \): \( \frac{729}{0.027} = 27000 \)
- For \( B_2 \): \( \frac{729}{0.216} = 3375 \)
- For \( B_3 \): \( \frac{729}{0.729} = 1 \)
Thus, the ratio becomes:
\[
27000 : 3375 : 1
\]
### Step 7: Reduce the ratio
Dividing each term by 1 gives:
\[
27000 : 3375 : 1
\]
This can be further simplified to:
\[
8 : 1 : \frac{1}{27}
\]
### Final Ratio
Thus, the final ratio of \( B_1 : B_2 : B_3 \) is:
\[
27 : 3.375 : 1
\]
