A charged particle moves with velocity `vec v = a hat i + d hat j` in a magnetic field `vec B = A hat i + D hat j.` The force acting on the particle has magnitude F. Then,

Options

A. (a)F = 0, if aD = dA
B. (b)F = 0, if aD = – dA
C. (c)F = 0, if aA = – dD
D. (d)`F prop (a^(2)+d^(2))^(1//2)xx(A^(2)+D^(2))^(1//2)`

Explanation

To solve the problem, we need to find the relationship between the force acting on a charged particle moving in a magnetic field, given its velocity and the magnetic field vectors.

### Step-by-Step Solution:

1. **Identify the Given Vectors**:
- The velocity vector of the charged particle is given as:
\[ \vec{v} = a \hat{i} + d \hat{j} \]
- The magnetic field vector is given as:
\[ \vec{B} = A \hat{i} + D \hat{j} \]

2. **Use the Formula for Magnetic Force**:
- The force \(\vec{F}\) acting on a charged particle moving in a magnetic field is given by the equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
- Here, \(q\) is the charge of the particle.

3. **Calculate the Cross Product \(\vec{v} \times \vec{B}\)**:
- To find the cross product, we can use the determinant form:
\[ \vec{v} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a & d & 0 \\ A & D & 0 \end{vmatrix} \]
- Expanding the determinant, we get:
\[ \vec{v} \times \vec{B} = \hat{i}(d \cdot 0 - 0 \cdot D) - \hat{j}(a \cdot 0 - 0 \cdot A) + \hat{k}(aD - dA) \]
- This simplifies to:
\[ \vec{v} \times \vec{B} = (aD - dA) \hat{k} \]

4. **Substitute Back into the Force Equation**:
- Now substituting the cross product back into the force equation:
\[ \vec{F} = q (aD - dA) \hat{k} \]

5. **Determine the Magnitude of the Force**:
- The magnitude of the force is given by:
\[ F = |q| |aD - dA| \]
- Thus, the force acting on the particle has a magnitude \(F\) which is proportional to the absolute value of the expression \(aD - dA\).

6. **Condition for Zero Force**:
- The force will be zero when:
\[ aD - dA = 0 \quad \Rightarrow \quad aD = dA \]
- This implies that the force acting on the particle is zero under this condition.

### Final Relation:
The relationship between the charge, velocity, magnetic field, and force can be summarized as:
\[ F = |q| |aD - dA| \]
The force is zero when \(aD = dA\).