Projectile Motion in an Electric Field

An electric field can be defined as the electrical properties present at a point in a space where a charged particle is present. An electric field exerts a force on other charged objects present in its vicinity.

The exerted force can be either attractive or repulsive depending on the type of charges which are interacting with each other. 

Projectile motion is referred to as the form of motion endured by a particle that is projected close to the surface of the earth and moves in a curved parabolic path only under the effect of gravity. 

Having understood what electric field and projectile motion are, now let’s move ahead and understand how the projectile motion takes place in an electric field.

Let us suppose that an electric field E is directed in space in the downward direction, and a charged particle of mass m, carrying charge q, and moving with velocity V is directed along the electrical field. Now, after entering into the electric field, the charge will experience a force F in the downward direction. 

The force experienced by the charged particle in the downward or in the y-direction (Fy) is equal to qE where q is the charge of the particle and E is the electric field.

The equation for the acceleration of a charged particle undergoing projectile motion in an electric field

The acceleration of a charged particle undergoing projectile motion in an electric field has two components, one in the x and the other in the y-direction. 

The acceleration in x-direction, ax=0 as the velocity is constant.           

The acceleration in y-direction = ay = F/m

Above, we have noted that F = qE

So, the acceleration in y-direction will be 

ay = qE/m

Here, q = Charge of the moving particle

          m = mass of the moving particle

          E = Electric field of the particle

The equation for initial velocity and distance travelled by the charged particle undergoing projectile motion in an electric field

The velocity V with which the charged particle is moving has two components, one in the x-direction and the other one in the y-direction.

The initial velocities in both directions areas are:- 

Initial velocity in x-direction (Ux) = V

That is, the velocity remains constant in the x-direction.  

Initial velocity in y-direction (Uy) = 0 

Now, the distance travelled by the charged particle in the x-direction will be given by:-

Sy = ut + ½ a(x)t² 

Above, we have deduced that a(x) = 0 

So, Sx = Vt + ½ at²

Suppose, charged particle covers x distance in X space. i.e. Sx = X

Then, distance in x-direction will be X = Vt

Time taken to cover distance (X) :-

t = X/V

Now, the distance travelled by the charged particle in the y-direction will be given by:-

Sy = ut + ½ ayt² 

Velocity in y-direction = 0 and acceleration in y-direction, ay = qE/m

So, Sy = ½ayt²

Suppose the charged particle covers a distance in Y space, i.e. Sy = y

Then, distance in y-direction will be y = ½ qet²/m 

In the above, we obtained that time taken by charged particle to cover the distance (X):- t = X/V

Replacing it, we get the distance covered in the y-direction to be y = ½qe(x²/V²) / m

The equation for the final velocity of a charged particle undergoing projectile motion in an electric field

The final velocity of the charged particle in x-direction and y-direction are respectively denoted by Vx and Vy. The equation for them are as follows:-

Equation for final velocity in x-direction, Vx = Ux + axt

Here, Ux = V and,

ax = 0

Thus, final velocity in x-direction,

Vx = V 

Similarly, equation for final velocity in y-direction, Vy = Uy + ayt

Here, Uy = 0 and,

a(y) = qE/m

Thus, final velocity in y-direction,

Vy = qEt/m

Above, we noted that t = X/V

So, Vy = qEX/mV

Deflection of the charged particle

Deflection of the charged particle is defined as the angle with which it is deflected from its original path. It is simply the ratio of velocity in the y-direction to velocity in the x-direction. It is denoted by tan Θ. 

The mathematical equation for it is as given below:-

tan Θ = Vy/Vx

tan Θ = qEX/mV²