Two electrons move parallel to each other with equal speed `'V'` the ratio of magnetic `&` electric force between them is

Options

A. `V//C`
B. `C//V`
C. `V^(2)//C^(2)`
D. `C^(2)//V^(2)`

Explanation

To find the ratio of the magnetic force to the electric force between two electrons moving parallel to each other with equal speed \( V \), we can follow these steps:

### Step 1: Determine the Electric Force
The electric force \( F_e \) between two electrons 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 an electron, and \( r \) is the distance between the two electrons.

### Step 2: Determine the Magnetic Field Due to One Electron
When an electron moves, it creates a magnetic field around it. The magnetic field \( B \) at a distance \( r \) from a long straight current-carrying wire is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]
where \( I \) is the current. For an electron moving with speed \( V \), the current \( I \) can be expressed as:
\[ I = \frac{e}{\Delta t} \]
where \( \Delta t \) is the time interval. The magnetic field due to one electron at the position of the other electron is:
\[ B = \frac{\mu_0 \cdot \frac{e}{\Delta t}}{2\pi r} \]

### Step 3: Calculate the Magnetic Force on One Electron
The magnetic force \( F_m \) on a charge \( q \) moving with velocity \( V \) in a magnetic field \( B \) is given by:
\[ F_m = qV \times B \]
For an electron, substituting \( q = e \) and \( B \):
\[ F_m = eV \cdot B = eV \cdot \frac{\mu_0 \cdot \frac{e}{\Delta t}}{2\pi r} \]

### Step 4: Substitute and Simplify
Substituting the expression for \( B \):
\[ F_m = eV \cdot \frac{\mu_0 \cdot \frac{e}{\Delta t}}{2\pi r} = \frac{\mu_0 e^2 V}{2\pi r \Delta t} \]

### Step 5: Calculate the Ratio of Magnetic Force to Electric Force
Now, we can find the ratio of the magnetic force \( F_m \) to the electric force \( F_e \):
\[ \text{Ratio} = \frac{F_m}{F_e} = \frac{\frac{\mu_0 e^2 V}{2\pi r \Delta t}}{\frac{1}{4\pi \epsilon_0} \cdot \frac{e^2}{r^2}} \]
This simplifies to:
\[ \text{Ratio} = \frac{\mu_0 \cdot 4\pi \epsilon_0 \cdot V \cdot r}{2\pi \Delta t} \]

### Step 6: Use the Relationship Between \( \mu_0 \) and \( \epsilon_0 \)
We know that:
\[ \mu_0 \epsilon_0 = \frac{1}{c^2} \]
where \( c \) is the speed of light. Thus, we can express the ratio as:
\[ \text{Ratio} = \frac{4V}{2c^2} = \frac{2V}{c^2} \]

### Final Result
The ratio of the magnetic force to the electric force between the two electrons is:
\[ \text{Ratio} = \frac{V}{c^2} \]