A proton travels in the x direction and enters a region where a constant uniform electric field is directed in the y direction. Derive an equation which describes the x and y cooedinates of the proton in the field as a function of time t.
To derive the equation describing the x and y coordinates of the proton in the electric field as a function of time, we'll make use of Newton's second law and the equation of motion in the presence of a constant uniform electric field.
1. Newton's Second Law:
The net force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, it can be expressed as: F = ma, where F is the net force, m is the mass, and a is the acceleration.
2. Equation of Motion in a Constant Electric Field:
In the presence of a constant uniform electric field, the force experienced by a charged particle is given by: F = qE, where F is the force, q is the charge of the particle, and E is the electric field vector.
Now, let's consider the proton's motion in the given scenario.
As mentioned, the proton travels in the x direction and experiences a constant uniform electric field directed in the y direction. Therefore, the only force acting on the proton is the electric force in the y direction, given by: F = qE.
According to Newton's second law, the force is equal to the mass of the particle multiplied by its acceleration. Since the proton's mass remains constant, we can rewrite the equation as follows: ma = qE.
Since the electric field vector is directed in the y direction, the only acceleration experienced by the proton is in the y direction. Hence, we can rewrite the equation as: may = qE.
The acceleration in the x direction is equal to zero since no force is acting in that direction. Therefore, ax = 0.
Now, we integrate the equations of motion to derive the equations for the x and y coordinates of the proton in the field as a function of time t.
Integrating ax = 0, we obtain: vx = u, where u is the initial velocity of the proton in the x direction.
Integrating may = qE, we obtain: vy = ut + c, where c is a constant of integration.
Since the proton starts at rest in the y direction, vy = 0 when t = 0. Substituting these values into the equation, we get: 0 = 0 + c. Therefore, c = 0.
So, the equation for the x coordinate is given by: x = ut.
And the equation for the y coordinate is given by: y = (1/2)at^2 = (1/2)(qE/m)t^2.
Therefore, the derived equation that describes the x and y coordinates of the proton in the electric field as a function of time t are:
x = ut,
y = (1/2)(qE/m)t^2.