Periodic motion, electric charge

In summary, periodic motion is a type of motion where an object repeats the same pattern of movement, characterized by a constant period. An electric charge is a fundamental property of matter that can affect periodic motion by causing attraction or repulsion between objects. Frequency is the number of cycles of periodic motion per second and is inversely proportional to the period. Periodic motion is related to the concept of equilibrium as the object moves back and forth around a central point where forces are balanced.
  • #1
tony873004
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Three parallel, infinite, uniformly charged planes are arranged as shown in Figure 24.32.
(it looks just like it's described. The middle plate is positive sigma, the outer plates are both negative sigma.)


Homework Statement



A small hole passes through the middle plane. At t=0 an electron emerges from the hole moving perpendicular to the planes with speed vo . Assuming vo is small enough that the electron does not collide with the negative plate, show that its motion is periodic, and find the period.


Homework Equations





The Attempt at a Solution



It's obvious that it will be periodic since there is no force to pull it off the x-axis, and there is no friction to rob it of energy.

I have a feeling it's like a pendulum, where the amplitude does not affect the period. But we haven't had any examples of periodic motion yet in E&M.

From the pendulum, we have a small amplitude formula for period of T=2pi sqrt(L/g). I imagine charge is somewhat analogus to gravity, and perhaps distance between the plates is somewhat analgous to L (length of pendulum). Where do I begin?



Homework Equations





The Attempt at a Solution


 
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  • #2
Using Gauss's theorem calculate the electric field between the plates. Then find force and acceleration of the electron. In SHM, acceleration = w^2.A, where A is the amplitude of SHM and that is the distance between the plates. From w you can find the period of the SHM.
 
  • #3


To begin, we can use the equation for the force on a particle due to an electric field, which is F = qE, where q is the charge of the particle and E is the electric field. In this case, the electric field is uniform and perpendicular to the planes, so the force on the electron will be constant and in the direction of the electric field.

Next, we can use Newton's second law, F = ma, to relate the force on the electron to its acceleration. Since the electron is moving perpendicular to the planes, the acceleration will also be perpendicular to the planes.

Now, we can use the equation for uniform circular motion, a = v^2/r, where v is the velocity of the electron and r is the radius of the circular path. Since the electron is moving in a circular path, we can relate its acceleration to its velocity and radius.

Finally, we can use the equation for the period of uniform circular motion, T = 2πr/v, to find the period of the electron's motion. This equation relates the radius of the circular path, velocity, and period.

Therefore, the period of the electron's motion can be calculated using the equations:

F = qE

F = ma

a = v^2/r

T = 2πr/v

By substituting and rearranging these equations, we can find that the period of the electron's motion is:

T = 2πm/qE

where m is the mass of the electron and q is its charge. This shows that the period is independent of the electron's initial velocity, as long as it is small enough to not collide with the negative plate.
 

Related to Periodic motion, electric charge

1. What is periodic motion?

Periodic motion is a type of motion in which an object repeats the same pattern of movement over and over again. This type of motion is characterized by a constant period, or amount of time it takes to complete one cycle of motion.

2. What is an electric charge?

An electric charge is a fundamental property of matter that describes the amount of electricity an object possesses. It can be either positive or negative, and like charges repel each other while opposite charges attract each other.

3. How does electric charge affect periodic motion?

Electric charge can affect periodic motion in a number of ways. For example, electric charge can cause objects to be attracted or repelled from one another, which can alter the path or speed of their periodic motion. It can also be used to create an electric current, which can power devices that rely on periodic motion.

4. What is the relationship between periodic motion and frequency?

Frequency is the number of cycles of periodic motion that occur in one second. Therefore, the higher the frequency, the faster the object is moving and completing its cycles. The period of the motion is inversely proportional to the frequency, meaning that as the frequency increases, the period decreases.

5. How is periodic motion related to the concept of equilibrium?

In periodic motion, an object moves back and forth around a central point or equilibrium position. This means that the object is in a state of balance, with equal forces acting on it in opposite directions. The object will continue to oscillate around this equilibrium point as long as the forces remain balanced.

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