Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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2g30001     Acceleration of an alpha particle by charged plates     2g30001
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An alpha particle is accelerated by two closely spaced, oppositely charged plates, as shown below.

Figure: An alpha particle moving between oppositely charged plates.

The alpha particle has a speed of m/s when it enters a slit in the positively charged plate. After traveling for 1 mm, it passes through a slit in the negatively charged plate. If the magnitude of the charge of each plate is , and if each plate has an area of , what will be the speed of the alpha particle when it reaches the negatively charged plate? (Note: the plate separation is small compared to the dimensions of the plates.)




Define a coordinate system with the +x axis in the direction of the velocity of the alpha particle. Define two points and along the path from the positively charged plate to the negatively charged plate.

Figure: Charged plates with coordinate system, initial and final points, and the constant electric field.

Let's begin by converting all units to m, kg, s, C, and combinations thereof. The charge on each plate has a magnitude . The area of each plate is . The plate separation is 1 mm, or 0.001 m.

Choose the Fundamental Principle that can be used to solve this problem–Conservation of Energy. The system of plates and particle are a closed system. Thus, the change in the total energy of the system is zero.

Because the force on the alpha particle by the electric field is in the same direction as the velocity of the particle, then the particle will speed up as it travels toward the negatively charged plate. As a result, its kinetic energy increases. Conservation of Energy tells us that this must be accompanied by a decrease in potential energy. It is the loss of electric potential energy that results in a gain in kinetic energy. Conceptually, we think of this as

If we calculate the initial kinetic energy and the change in electric potential energy, then the final kinetic energy and final speed can be determined.

The alpha particle is a helium nucleus with 2 neutrons and 2 protons; it's atomic mass is 4 g/mol which is . Its initial kinetic energy is

The change in potential energy of the alpha particle (that has a charge of ) from point to point is:

Thus, the final kinetic energy is

From this, calculate the final speed, using .