The change in the kinetic energy of a charged particle moving in a uniform magnetic field will be zero when its velocity is

Options

A. Parallel to the magnetic field
B. Perpendicular to the magnetic field
C. At any angle with the magnetic field
D. All of these

Answer

To solve the problem of when the change in kinetic energy of a charged particle moving in a uniform magnetic field will be zero, we can follow these steps:

### Step 1: Understand the Concept of Kinetic Energy
The kinetic energy (KE) of a charged particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 2: Analyze the Motion in a Magnetic Field
When a charged particle moves in a uniform magnetic field, it experiences a magnetic force that acts perpendicular to its velocity. This force does not do work on the particle because it is always perpendicular to the direction of motion. Therefore, the speed (and hence the kinetic energy) of the particle remains constant.

### Step 3: Relate Centripetal Force and Magnetic Force
The magnetic force acting on the charged particle can be expressed as:
\[ F = qvB \]
where \( q \) is the charge of the particle and \( B \) is the magnetic field strength. This magnetic force acts as the centripetal force required for circular motion, which can be expressed as:
\[ F = \frac{mv^2}{r} \]
where \( r \) is the radius of the circular path.

### Step 4: Set the Forces Equal
Setting the magnetic force equal to the centripetal force gives:
\[ qvB = \frac{mv^2}{r} \]
From this, we can derive the expression for velocity:
\[ v = \frac{qBr}{m} \]

### Step 5: Substitute Velocity into Kinetic Energy
Now, substituting this expression for \( v \) back into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\frac{qBr}{m}\right)^2 = \frac{q^2B^2r^2}{2m} \]
This shows that the kinetic energy depends on the charge, magnetic field, radius, and mass, but not on the velocity itself.

### Step 6: Conclusion on Change in Kinetic Energy
Since the kinetic energy derived does not depend on the velocity of the particle, the change in kinetic energy will be zero regardless of the direction of the velocity (whether it is parallel, perpendicular, or at any angle to the magnetic field).

### Final Answer
The change in kinetic energy of a charged particle moving in a uniform magnetic field will be zero when its velocity is at any angle to the magnetic field. Therefore, the correct answer is:
**All of the above options are correct.**

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