Under what condition is the force acting on a charge moving through a uniform magnetic field minimum ?

Options

A. `theta=180^@`
B. `theta=0^@` or `90^@`
C. `theta=90^@`
D. `theta=0^@` or `180^@`

Answer

To find the condition under which the force acting on a charge moving through a uniform magnetic field is minimum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Force on a Charge in a Magnetic Field**:
The force \( F \) acting on a charge \( q \) moving with a velocity \( v \) in a magnetic field \( B \) is given by the formula:
\[ F = q v B \sin \theta \]
where \( \theta \) is the angle between the velocity vector \( v \) and the magnetic field vector \( B \).

2. **Identify the Conditions for Minimum Force**:
To find the minimum force, we need to analyze the term \( \sin \theta \). The sine function varies between -1 and 1. The minimum value of \( \sin \theta \) is 0, which occurs when:
- \( \theta = 0^\circ \) (the charge moves parallel to the magnetic field)
- \( \theta = 180^\circ \) (the charge moves anti-parallel to the magnetic field)

3. **Calculate the Minimum Force**:
When \( \sin \theta = 0 \), the force \( F \) becomes:
\[ F = q v B \cdot 0 = 0 \]
Thus, the force acting on the charge is zero, which is indeed the minimum value.

4. **Conclusion**:
Therefore, the force acting on a charge moving through a uniform magnetic field is minimum (zero) when the angle \( \theta \) between the velocity of the charge and the magnetic field is either \( 0^\circ \) or \( 180^\circ \).

### Final Answer:
The force acting on a charge moving through a uniform magnetic field is minimum when \( \theta = 0^\circ \) or \( \theta = 180^\circ \).

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