If a charged particle is a plane perpendicular to a uniform magnetic field with a time period T Then

Options

A. `T^(2)propr^(3)`
B. `T^(2)propr`
C. `Tpropr^(2)`
D. `Tpropr^(0)`

Answer

To solve the problem step-by-step, we start by analyzing the motion of a charged particle in a magnetic field.

### Step-by-Step Solution:

1. **Understanding the Motion**:
A charged particle moving in a plane perpendicular to a uniform magnetic field will undergo circular motion due to the magnetic force acting on it.

2. **Applying Lorentz Force**:
The magnetic force (Lorentz force) acting on the charged particle is given by:
\[ F = q(v \times B) \]
Since the velocity \(v\) and magnetic field \(B\) are perpendicular, we can simplify this to:
\[ F = qvB \]

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

4. **Setting Forces Equal**:
Since the magnetic force provides the centripetal force, we can set these two expressions for force equal to each other:
\[ qvB = \frac{mv^2}{r} \]

5. **Cancelling \(v\)**:
We can cancel \(v\) from both sides (assuming \(v \neq 0\)):
\[ qB = \frac{mv}{r} \]

6. **Expressing Velocity**:
The velocity \(v\) can also be expressed in terms of the time period \(T\) of the circular motion. The distance traveled in one complete circle (circumference) is \(2\pi r\), so:
\[ v = \frac{2\pi r}{T} \]

7. **Substituting Velocity**:
Substituting this expression for \(v\) back into the equation gives:
\[ qB = \frac{m(2\pi r/T)}{r} \]
Simplifying this, we get:
\[ qB = \frac{2\pi m}{T} \]

8. **Solving for Time Period \(T\)**:
Rearranging the equation to solve for \(T\):
\[ T = \frac{2\pi m}{qB} \]

9. **Analyzing the Relationship**:
From the equation \(T = \frac{2\pi m}{qB}\), we see that \(T\) is independent of \(r\) (the radius of the circular path). This means:
\[ T \propto r^0 \]

### Conclusion:
Thus, the correct answer is that the time period \(T\) is proportional to \(r^0\), which corresponds to option D.