Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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(N)=# of solutions
15d0001     Speed of a proton and electron interacting with a charged sphere.     15d0001
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Question

 

The charged sphere shown below has a net positive charge of 1 nC and is very massive compared to a single proton.


Figure: A charged sphere.

  1. What is the speed of a proton at point 2 if it is moving to the right with a speed of at point 1?
  2. What is the speed of a proton at point 2 if it is moving to the left with a speed of at point 1?
  3. What is the speed of an electron at point 2 if it is moving to the right with a speed of at point 1?

 



Solution

 

(a) Apply the energy principle. Define the system as the charged sphere and the proton. Treat them both as point particles, meaning that they only have rest energy and kinetic energy. And of course, there is energy due to their electrical interaction, which is called electrical potential energy or Coulomb potential energy. Also, define the initial and final states of the system. In this case, the initial state is when the proton is at point 1, and the final state is when the proton is at point 2. The energy principle states that

Substitute the energy of the system which is the sum of the particle energies and their interaction energy.

Write the particle energy as the sum of its rest energy and kinetic energy. Note that rest energy cancels out of both sides of the equation, if we assume that the mass of the proton and sphere haven't changed during the process.

It is an isolated system, with no external forces on the system. Thus, the work done on the system is zero. Substitute the work done on the system into the energy principle.

Substitute expressions for kinetic energy and elecitational potential energy. Note that the sphere has a very large mass compared to the proton. As a result, it essentially remains at rest during the interaction. Thus, the kinetic energy of the system is simply the kinetic energy of the proton.

Substitute the initial speed and initial and final distances of the proton from the sphere, and solve for the final kinetic energy of the proton (at point 2).

The first term in the above equation is the initial kinetic energy of the proton.

The second term is the negative of the change in electric potential energy of the system.

The final kinetic energy of the proton is thus

Now, solve for the final speed of the proton (at point 2).

The final speed of the proton (at point 2) is greater than the initial speed of the proton (at point 1). This make sense because the proton and sphere repel one another. Since the force on the proton is in the same direction as its displacement (from point 1 to point 2), the sphere does positive work on the proton thus making it speed up. Or, in thinking of the proton-sphere system, as the proton moves from point 1 to point 2, the system loses electrical potential energy. As a result, it gains kinetic energy and the proton speeds up.

Also note that the proton's speed is much less than the speed of light (roughly one-thousandth of the speed of light). Therefore, we are justified in using the approximate, non-classical expression for kinetic energy, , in the solution.

(b) If the proton is moving to the left at point 1, it has the same kinetic energy as if it is moving to the right. Kinetic energy depends on speed, not the direction of the velocity. As a result, whether the proton moves to the left or right at point 1, if its initial speed at point 1 is , then its speed at point 2 is .

(c) If the particle is an electron (instead of a proton), then its charge is . This changes the sign on the electric potential energy of the system. Also, its mass is . Again, the initial state of the system is when the electron is at point 1. The final state is when the electron is at point 2. Since the initial velocity of the electron is to the right and since the force on the proton is to the left, we expect the electron to slow down. Applying the energy principle to the sphere-electron system gives

The first term is the initial kinetic energy of the electron.

The second term is the negative of the change in electric potential energy of the system.

The final kinetic energy of the electron is thus

Note that because of the electron's smaller mass, its initial kinetic energy is much smaller than in part (a) for the moving proton. Yet, the change in electrical potential energy is the same, except for negative sign because of the negative charge of the electron. In this case, the electron's initial kinetic energy is quite small compared to the change in electric potential energy as it moves from point 1 to point 2. As a result, the final kinetic energy of the electron is negative. BUT THIS CAN'T BE!

Kinetic energy can never be negative. There are two possibilities: (1) we made an algebraic or numeric error or (2) the electron never reaches point (2). That is, as it moves away from point 1 toward point 2, it slows down. Eventually (if it doesn't escape), it reaches zero speed, changes direction, and accelerates toward the sphere. In this case, the electron must have reached its turning point (zero kinetic energy) before reaching point 2. Let's calculate its final distance at the turning point.

This means that at its initial distance of 0.01 m from the sphere, the electron is very near to its turning point. It only travels another 0.0005 m or so before turning around. But note that this is not significant because we cannot calculate the final position of the electron to this many significant figures. It's certainly an interesting result, and one that we may not have expected until doing the calculation.