Uniform electric and magnetic fields are applied along the same direction and a negatively charged particle is given an initial velocity at an acute angle with the direction of the fields. What kind of a path will the particle follow?

Options

A. Helical path with an increasing pitch.
B. Helical path with a decreasing pitch.
C. Helical path with an increasing pitch followed by a decreasing pitch.
D. Helical path with a decreasing pitch followed by an increasing pitch.

Explanation

To solve the problem of the path followed by a negatively charged particle in uniform electric and magnetic fields applied in the same direction, we can break down the solution into several steps:

### Step 1: Identify Forces Acting on the Particle
- The negatively charged particle experiences two forces: the electric force (\(F_e\)) and the magnetic force (\(F_B\)).
- The electric force is given by \(F_e = qE\), where \(q\) is the charge of the particle (negative in this case) and \(E\) is the electric field strength. The direction of this force will be opposite to the direction of the electric field due to the negative charge.
- The magnetic force is given by \(F_B = q(v \times B)\), where \(v\) is the velocity of the particle and \(B\) is the magnetic field strength. The direction of this force is determined by the right-hand rule, but since the charge is negative, the force will act in the opposite direction.

### Step 2: Break Down the Velocity Components
- The initial velocity of the particle is at an acute angle with respect to the direction of the fields. We can decompose this velocity into two components:
- \(v_x = v \cos \theta\) (component along the direction of the fields)
- \(v_y = v \sin \theta\) (component perpendicular to the direction of the fields)

### Step 3: Analyze the Effects of Each Force
- The electric force will act along the direction of the electric field, affecting the \(v_x\) component. Since the electric force is opposite to the field direction for a negative charge, it will decelerate the particle in the \(x\)-direction.
- The magnetic force, acting on the \(v_y\) component, will cause the particle to move in a circular path in the \(y-z\) plane, as it is perpendicular to the velocity component.

### Step 4: Determine the Path of the Particle
- The particle will undergo circular motion due to the magnetic force while simultaneously being affected by the electric force in the \(x\)-direction.
- As the particle moves, the \(v_x\) component will decrease due to the electric force, while the \(v_y\) component will continue to cause circular motion. This results in a helical path.

### Step 5: Analyze the Pitch of the Helical Path
- The pitch of the helix is determined by the \(v_x\) component. As \(v_x\) decreases, the distance traveled in the \(x\)-direction per cycle will also decrease, leading to a decreasing pitch initially.
- However, as the particle continues to move, the \(v_x\) component will eventually stabilize, and the pitch may start to increase again, leading to a helical path with an increasing pitch.

### Conclusion
The particle will follow a helical path with a decreasing pitch initially, followed by an increasing pitch as it continues to move through the fields.