In which of the following situations, the magnetic field can accelerate a charge particle at rest?
I. When the magnetic field is uniform with respect to time as well as position
II. When the magnetic field is time varying but uniform w.r.t. position
III. When the magnetic field is time independent but position dependent

Options

A. I, II & II
B. lll only
C. ll only
D. None of these

Answer

To determine in which of the given situations a magnetic field can accelerate a charged particle at rest, we need to analyze each statement based on the principles of electromagnetic induction.

### Step-by-Step Solution:

1. **Understanding the Concept of Induced EMF**:
- According to Faraday's law of electromagnetic induction, an electromotive force (EMF) is induced in a circuit when there is a change in magnetic flux through the circuit. The induced EMF (E) can be calculated using the formula:
\[ E = -\frac{d\Phi}{dt} \]
- Where \(\Phi\) is the magnetic flux, given by:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]

2. **Analyzing Statement I**:
- **Statement**: When the magnetic field is uniform with respect to time as well as position.
- **Analysis**: If the magnetic field (B) is uniform and constant in both time and position, then there is no change in magnetic flux (\(d\Phi/dt = 0\)). Therefore, the induced EMF is zero, which means no current is produced and the charged particle cannot be accelerated.
- **Conclusion**: This statement is **incorrect**.

3. **Analyzing Statement II**:
- **Statement**: When the magnetic field is time varying but uniform with respect to position.
- **Analysis**: If the magnetic field varies with time (i.e., \(B\) is a function of time), then \(dB/dt \neq 0\). Since the area (A) is constant, we can take it outside the derivative:
\[ E = -A \frac{dB}{dt} \]
- Here, a non-zero EMF is induced, which can produce a current and thus accelerate the charged particle at rest.
- **Conclusion**: This statement is **correct**.

4. **Analyzing Statement III**:
- **Statement**: When the magnetic field is time independent but position dependent.
- **Analysis**: If the magnetic field is time-independent, then \(dB/dt = 0\). This means that the magnetic flux does not change with time, leading to zero induced EMF. Therefore, no current is produced, and the charged particle cannot be accelerated.
- **Conclusion**: This statement is **incorrect**.

### Final Conclusion:
Only **Statement II** is correct. The magnetic field can accelerate a charged particle at rest only when the magnetic field is time varying but uniform with respect to position.

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