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**Motion of Charged Particles in a Uniform Electric Field**

Montwood High School AP Physics C R. Casao

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**Acceleration in Uniform Electric Field**

The motion of a charged particle in a uniform electric field is equivalent to that of a projectile moving in a uniform gravitational field. When a charge q is placed in an electric field E, the electric force on the charge is F = E·q. From Newton’s second law, F = m·a, therefore, m·a = E·q. The acceleration of the charge is:

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**Acceleration in Uniform Electric Field**

If E is uniform (constant in magnitude and direction), then the acceleration is constant. If the charge is positive, the acceleration will be in the direction of the electric field. If the charge is negative, the acceleration will be in the direction opposite the electric field.

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**Acceleration in Uniform Electric Field**

The electric field in the region between two oppositely charged flat metal plates is considered to be uniform. If an electron is projected horizontally into an electric field with an initial velocity vo, it will be accelerated by the electric field.

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**Acceleration in Uniform Electric Field**

The acceleration will be in the positive y direction (the direction of the electric field). Because the acceleration is constant, we can apply the two-dimensional kinematics equations for projectile motion: vo = vx = constant (no acceleration in the horizontal direction)

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**Acceleration in Uniform Electric Field**

Final vertical speed: vyf = vyi + (a·t); initial velocity in y-direction is zero because the electron enters the field horizontally. Vertical displacement: Dy = (vyi·t) + (0.5·a·t2) Horizontal displacement: x = vx·t The time t that the electron is accelerating vertically within the electric field is equal to the time during which it is traveling horizontally through the electric field.

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**Acceleration in Uniform Electric Field**

Once the electron leaves the uniform electric field, it continues to move in a straight line with a speed greater than its original speed. The angle at which the electron exits the electric field is given by:

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**Forces on electron beam in a TV tube (CRT) **

F = Q E and F = m g (vector equations) 24-22 Forces in a TV tube

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**TV tube with electron-deflecting charged plates (orange)**

F = Q E

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What About Gravity? The gravitational force acting on the mass of the electron has been neglected because the magnitude of this force is 9.11 x kg ·9.8 m/s2 = x N, which is small in comparison to the electric force acting on the electron. The same is true for a proton.

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