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, we need to analyze the motion of an electron moving in a circular path under the influence of a magnetic field. Here’s a step-by-step solution:

### Step 1: Understand the Motion of the Electron
The electron is moving in a circular path due to the magnetic force acting on it. The magnetic force provides the necessary centripetal force to keep the electron in circular motion.

**Hint:** Recall that the magnetic force acting on a charged particle moving in a magnetic field is given by the Lorentz force equation.

### Step 2: Identify the Forces Acting on the Electron
The magnetic force \( F \) acting on the electron can be expressed as:
\[ F = e \cdot v \cdot B \]
where:
- \( e \) is the charge of the electron,
- \( v \) is the velocity of the electron,
- \( B \) is the magnetic field strength.

**Hint:** Remember that the direction of the magnetic force can be determined using the right-hand rule or Fleming's left-hand rule.

### Step 3: Relate the Magnetic Force to Centripetal Force
For circular motion, the magnetic force acts as the centripetal force:
\[ F = \frac{m v^2}{r} \]
Setting the two expressions for force equal gives:
\[ e \cdot v \cdot B = \frac{m v^2}{r} \]

**Hint:** This equation shows the balance between the magnetic force and the centripetal force.

### Step 4: Analyze the Work Done on the Electron
Since the magnetic force is always perpendicular to the velocity of the electron, the work done by the magnetic field on the electron is zero. Work done is defined as:
\[ W = F \cdot d \cdot \cos(\theta) \]
where \( \theta \) is the angle between the force and displacement vectors. In this case, \( \theta = 90^\circ \), so:
\[ W = F \cdot d \cdot \cos(90^\circ) = 0 \]

**Hint:** The work-energy principle states that if no work is done, there is no change in kinetic energy.

### Step 5: Determine the Change in Kinetic Energy
Since the work done on the electron is zero, the change in kinetic energy \( \Delta KE \) is also zero:
\[ \Delta KE = 0 \]

**Hint:** If the kinetic energy does not change, then the energy acquired by the electron is also zero.

### Conclusion
The energy acquired by the electron after describing half a revolution in a magnetic field is:
\[ \text{Energy acquired} = 0 \]

Thus, the correct answer from the given options is **D: zero**.