Two protons move parallel to each other with an equal velocity `v=300kms^-1`. Find the ratio of forces of magnetic and electric interaction of the protons.

Explanation

To find the ratio of the forces of magnetic and electric interaction of two protons moving parallel to each other with a velocity \( v = 300 \, \text{km/s} \), we can follow these steps:

### Step 1: Identify the Electric Force
The electric force \( F_e \) between two protons can be calculated using Coulomb's law:
\[ F_e = \frac{k \cdot e^2}{r^2} \]
where \( k = \frac{1}{4\pi \epsilon_0} \) is Coulomb's constant, \( e \) is the charge of a proton (\( e \approx 1.6 \times 10^{-19} \, \text{C} \)), and \( r \) is the distance between the two protons.

### Step 2: Identify the Magnetic Force
The magnetic force \( F_m \) between two moving charges can be calculated using the formula:
\[ F_m = \frac{\mu_0}{4\pi} \cdot \frac{e^2 \cdot v^2}{r^2} \]
Here, \( \mu_0 \) is the permeability of free space.

### Step 3: Calculate the Ratio of Forces
To find the ratio of the magnetic force to the electric force, we can express it as:
\[ \frac{F_m}{F_e} = \frac{\frac{\mu_0}{4\pi} \cdot \frac{e^2 \cdot v^2}{r^2}}{\frac{1}{4\pi \epsilon_0} \cdot \frac{e^2}{r^2}} \]
This simplifies to:
\[ \frac{F_m}{F_e} = \frac{\mu_0 \cdot v^2}{\epsilon_0} \]

### Step 4: Use the Relationship Between \(\mu_0\), \(\epsilon_0\), and \(c\)
We know that:
\[ \mu_0 \epsilon_0 = \frac{1}{c^2} \]
where \( c \) is the speed of light in vacuum (\( c \approx 3 \times 10^8 \, \text{m/s} \)). Therefore, we can rewrite the ratio as:
\[ \frac{F_m}{F_e} = \frac{v^2}{c^2} \]

### Step 5: Substitute the Values
Substituting \( v = 300 \, \text{km/s} = 300 \times 10^3 \, \text{m/s} \):
\[ \frac{F_m}{F_e} = \left(\frac{300 \times 10^3}{3 \times 10^8}\right)^2 \]
Calculating this gives:
\[ \frac{F_m}{F_e} = \left(\frac{300}{3 \times 10^5}\right)^2 = \left(\frac{1}{1000}\right)^2 = \frac{1}{10^6} \]

### Conclusion
Thus, the ratio of the magnetic force to the electric force is:
\[ \frac{F_m}{F_e} = 1 : 10^6 \]