To solve the problem of finding the force of repulsion between two short bar magnets with given magnetic moments, we can follow these steps:
### Step 1: Understand the Given Data
We have two bar magnets with the following magnetic moments:
- \( M_1 = 400 \, \text{A cm}^2 \)
- \( M_2 = 800 \, \text{A cm}^2 \)
The distance between the centers of the two magnets is:
- \( R = 20 \, \text{cm} = 0.2 \, \text{m} \)
### Step 2: Convert Magnetic Moments to SI Units
To use the magnetic moments in calculations, we need to convert them from A cm² to A m²:
- \( M_1 = 400 \, \text{A cm}^2 = 400 \times 10^{-4} \, \text{A m}^2 = 0.04 \, \text{A m}^2 \)
- \( M_2 = 800 \, \text{A cm}^2 = 800 \times 10^{-4} \, \text{A m}^2 = 0.08 \, \text{A m}^2 \)
### Step 3: Use the Formula for Force of Repulsion
The formula for the force of repulsion \( F \) between two magnetic dipoles is given by:
\[
F = \frac{6 \mu_0 M_1 M_2}{4 \pi R^4}
\]
where \( \mu_0 \) (the permeability of free space) is approximately \( 4 \pi \times 10^{-7} \, \text{T m/A} \).
### Step 4: Substitute the Values into the Formula
Now we can substitute the values into the formula:
\[
F = \frac{6 \times (4 \pi \times 10^{-7}) \times (0.04) \times (0.08)}{4 \pi (0.2)^4}
\]
### Step 5: Simplify the Expression
The \( 4 \pi \) in the numerator and denominator cancels out:
\[
F = \frac{6 \times 10^{-7} \times 0.04 \times 0.08}{(0.2)^4}
\]
Calculating \( (0.2)^4 \):
\[
(0.2)^4 = 0.0016
\]
Now substituting this value back into the equation:
\[
F = \frac{6 \times 10^{-7} \times 0.0032}{0.0016}
\]
### Step 6: Calculate the Force
Now, calculate the force:
\[
F = \frac{1.92 \times 10^{-9}}{0.0016} = 1.2 \times 10^{-6} \, \text{N}
\]
### Step 7: Convert to Dyne
Since \( 1 \, \text{N} = 10^5 \, \text{dyne} \):
\[
F = 1.2 \times 10^{-6} \, \text{N} = 12 \, \text{dyne}
\]
### Final Answer
The force of repulsion between the two magnets is \( 12 \, \text{dyne} \).
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