An electron with mass m, velocity v and charge e describe half a revolution in a circle of radius r in a magnetic field B. It will acquire energy equal to

Options

A. `1//2mv^(2)`
B. `1//4mv^(2)`
C. `pirBev`
D. zero

Answer

To solve the problem of an electron moving in a circular path under the influence of a magnetic field, we will analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Motion**:
The electron is moving in a circular path due to the magnetic field. The force acting on the electron is the magnetic Lorentz force, which acts as the centripetal force required for circular motion.

2. **Centripetal Force**:
The centripetal force \( F_c \) required to keep the electron in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the circular path.

3. **Magnetic Force**:
The magnetic force acting on the electron can be expressed as:
\[ F_B = e v B \]
where \( e \) is the charge of the electron, \( v \) is its velocity, and \( B \) is the magnetic field strength.

4. **Equating Forces**:
For the electron to move in a circular path, the centripetal force must equal the magnetic force:
\[ \frac{mv^2}{r} = e v B \]
This equation confirms that the magnetic force provides the necessary centripetal force.

5. **Power and Work Done**:
The power developed by the centripetal force is given by:
\[ P = F \cdot v \]
Since the centripetal force is always perpendicular to the velocity of the electron, the angle between them is \( 90^\circ \). Therefore, the dot product becomes zero:
\[ P = F_c \cdot v \cdot \cos(90^\circ) = 0 \]
This means that the power developed by the centripetal force is zero.

6. **Change in Kinetic Energy**:
Since power is defined as the rate of change of kinetic energy, and we have established that the power is zero, we can conclude:
\[ \frac{dK_e}{dt} = 0 \implies \Delta K_e = 0 \]
This indicates that there is no change in the kinetic energy of the electron as it moves through half a revolution.

### Final Answer:
The energy acquired by the electron after describing half a revolution in the magnetic field is equal to **zero**.

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