Heading: Given Information

We are given the following information:
- Charge q with mass m is moving with velocity (V.i V.j).
- The charge enters a region with a magnetic field (B.j) and an electric field (E.j).

Heading: Pitch for 1st revolution

The pitch for the first revolution is the distance traveled by the charge in one complete revolution. To find the expression for the pitch, we need to consider the effects of both the magnetic field and the electric field on the charge.

Heading: Effect of Magnetic Field

When a charge moves in a magnetic field, it experiences a force perpendicular to its velocity. This force is given by the equation:

F = q(V x B)

Where:
- F is the force experienced by the charge
- q is the charge
- V is the velocity of the charge
- B is the magnetic field

Heading: Effect of Electric Field

When a charge moves in an electric field, it experiences a force in the direction of the electric field. This force is given by the equation:

F = qE

Where:
- F is the force experienced by the charge
- q is the charge
- E is the electric field

Heading: Equating the Forces

Since the charge is moving in both the magnetic field and the electric field simultaneously, the total force experienced by the charge is the sum of the forces due to the magnetic field and the electric field. We can equate these forces to find the expression for the pitch.

q(V x B) = qE

Heading: Simplifying the Equation

Let's simplify the equation by canceling out the charge q:

(V x B) = E

Since we want to find the pitch, we need to find the distance traveled by the charge in one complete revolution. This distance is given by the circumference of the circular path.

Heading: Distance Traveled in One Revolution

The distance traveled in one revolution is given by the formula:

Distance = 2πr

Where:
- Distance is the distance traveled in one revolution
- r is the radius of the circular path

Heading: Finding the Radius

To find the radius of the circular path, we need to consider the velocity of the charge and the magnetic field. The magnetic force provides the centripetal force required for the circular motion of the charge. The centripetal force is given by the equation:

F = mv^2/r

Where:
- F is the centripetal force
- m is the mass of the charge
- v is the velocity of the charge
- r is the radius of the circular path

Equating the centripetal force with the magnetic force, we have:

q(V x B) = mv^2/r

Simplifying the equation, we get:

r = mv/(qB)

Heading: Substituting the Values

Now, let's substitute the given values into the expression for the pitch:

Distance = 2πr = 2πmv/(qB)

Since the velocity is given as (V.i V.j), we need to find the dot product of the velocity and the electric field to get the expression for the pitch.

(V. E.) = VxE = VxEj = V.Ej

Substituting this into