A charged particle moves in a magnetic field `vecB=10hati` with initial velocity `vecu=5veci+4hatj`. The path of the particle will be

Options

A. (a)Straight line
B. (b)circle
C. (c)helical
D. (d)None of these

Explanation

To determine the path of a charged particle moving in a magnetic field, we can follow these steps:

### Step 1: Identify the given quantities
- Magnetic field \( \vec{B} = 10 \hat{i} \)
- Initial velocity \( \vec{u} = 5 \hat{i} + 4 \hat{j} \)

### Step 2: Analyze the components of the velocity
- The velocity can be broken down into its components:
- \( u_x = 5 \) (along the x-axis)
- \( u_y = 4 \) (along the y-axis)

### Step 3: Determine the effect of the magnetic field on the particle
- The magnetic force acting on a charged particle moving in a magnetic field is given by the Lorentz force:
\[ \vec{F} = q(\vec{u} \times \vec{B}) \]
where \( q \) is the charge of the particle.

### Step 4: Calculate the cross product \( \vec{u} \times \vec{B} \)
- The magnetic field \( \vec{B} \) is directed along the x-axis, and the velocity vector has components in both the x and y directions.
- The cross product can be calculated as follows:
\[ \vec{u} \times \vec{B} = (5 \hat{i} + 4 \hat{j}) \times (10 \hat{i}) \]
Since the cross product of any vector with itself is zero, we only need to consider the \( \hat{j} \) component:
\[ \vec{u} \times \vec{B} = 5 \hat{i} \times 10 \hat{i} + 4 \hat{j} \times 10 \hat{i} = 0 + 40 \hat{k} = 40 \hat{k} \]

### Step 5: Interpret the motion of the particle
- The \( \hat{k} \) component indicates that the force is acting in the z-direction, while the particle has a component of velocity in the x-direction (5) and y-direction (4).
- The particle will experience a circular motion in the plane perpendicular to the magnetic field (xy-plane) due to the magnetic force, while it continues to move in the z-direction due to the initial velocity.

### Step 6: Conclude the path of the particle
- Since the particle moves in a circular path in the xy-plane while simultaneously moving linearly in the z-direction, the overall path of the particle will be helical.

### Final Answer
The path of the particle will be helical.

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