In a homogeneous magnetic field B there is an electron moving in a circular orbit with a speed v. Find the ratio of the magnetic field generated by the moving electron at the middle of the circle and the magnetic field making it revolve.

Answer

To solve the problem, we need to find the ratio of the magnetic field generated by a moving electron at the center of its circular path (denoted as \( B_E \)) to the external magnetic field \( B \) that causes the electron to revolve in that circular path.

### Step-by-Step Solution:

1. **Understanding the Current Due to the Electron:**
- An electron moving in a circular path can be treated as a current loop. The current \( I \) due to the electron can be defined as the charge passing through a point per unit time.
- The charge of an electron is denoted as \( e \).

2. **Calculating the Current \( I \):**
- The time period \( T \) for one complete revolution is given by the distance traveled (circumference of the circle) divided by the speed \( v \):
\[ T = \frac{2\pi R}{v} \]
- The current \( I \) can be expressed as:
\[ I = \frac{e}{T} = \frac{e}{\frac{2\pi R}{v}} = \frac{ev}{2\pi R} \]

3. **Finding the Magnetic Field \( B_E \) at the Center of the Circular Path:**
- The magnetic field at the center of a circular loop carrying current \( I \) is given by:
\[ B_E = \frac{\mu_0 I}{2R} \]
- Substituting the expression for \( I \):
\[ B_E = \frac{\mu_0 \left(\frac{ev}{2\pi R}\right)}{2R} = \frac{\mu_0 ev}{4\pi R^2} \]

4. **Magnetic Field \( B \) Causing the Electron to Revolve:**
- The magnetic force acting on the electron moving in the magnetic field \( B \) provides the centripetal force necessary for circular motion. This force is given by:
\[ F = e v B \]
- The centripetal force required for circular motion is:
\[ F = \frac{mv^2}{R} \]
- Setting these two forces equal gives:
\[ evB = \frac{mv^2}{R} \]
- Rearranging for \( B \):
\[ B = \frac{mv}{eR} \]

5. **Calculating the Ratio \( \frac{B_E}{B} \):**
- Now, we can find the ratio of the magnetic fields:
\[ \frac{B_E}{B} = \frac{\frac{\mu_0 ev}{4\pi R^2}}{\frac{mv}{eR}} = \frac{\mu_0 e^2}{4\pi m R} \]

### Final Result:
The ratio of the magnetic field generated by the moving electron at the center of the circle to the magnetic field causing it to revolve is:
\[ \frac{B_E}{B} = \frac{\mu_0 e^2}{4\pi m R} \]