Answer :

Given,

d_{1},d_{2} are the separations between capacitor plates in the upper and lower capacitors respectively.

a is the length of each plate

Area of each plates a^{2}

S_{y} is the distance that the electron must travel in order to avoid collision in Y-direction d_{1}/2

S_{x} is the distance that the electron must travel in order to avoid collision in X-direction a

V is the potential difference between the given series arrangement of capacitors.

e is the charge of electron released in between the plates

**Formula used**

In order to avoid a collision with plates, the electron should have an initial velocity, v. Hence, with ‘v’ velocity, the electron should travel a distance of ‘d_{1}/2’ in Y-direction and ‘a’ in X-direction

Since the electric field is acting only in Y-direction, the electron will travel with constant velocity, v, in X-direction.

Hene the external force, neglecting gravitational and other forces, acting on the electron is the force due to the electric fieldqE). Hence, according to Newton’s second law of motion, we can write,

Where,

mmass of electron;

a_{y} acceleration of electron in Y-direction;

q=e=charge of electron;

E= Magnitude of Electric field acting between the plates of capacitor.

We know that the distance that must be traveled in X-directiona

So, by the equations of motion, this can be represented as,

Or,

Where,

t time taken to travel ‘a’ distance

Acceleration in X-direction is Zero)

And the distance that must be traveled in Y-directiond_{1}/2

Hence, by the equation of motion, assuming no initial velocity in Y-direction as the electron is projected horizontally.

From eqn.1 and eqn.2, eqn. 3 can be modified as,

Now, let C_{1} and C_{2} be the capacitance of the upper and lower capacitors. Hence the effective capacitance, C_{eff} of the series arrangement is,

Or,

Where,

and

ε_{0} Permittivity of free space, in between the capacitor plates.

Hence,

Or,

Now, the magnitude of electric field, E, in the upper capacitor is given by,

Where, V_{1} Potential difference in the upper capacitor and is equal to,

Where,

Q= charge in each capacitor total charge in the arrangement, since it is a series arrangement

Hence, Q can be calculated as,

Where V total potential difference

Or,

Substituting the above equation and the value of C1 in eqn.6, we get,

Or,

Substituting the above expression in eqn.5, we get,

Or,

Substituting the above expression in eqn.4, we get

By re-arranging,

The above expression is the least value of horizontal initial velocity needed for the electron to cross the capacitor plates without collision.

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