Question

Show that an electron in a one-dimensional electri. cal potential \(V(x)=A x^{2},\) where the constant \(A\) is a positive real number, will execute simple harmonic motion about the origin. What is the period of that motion?

Short answer

Question: Show that an electron in a one-dimensional electric potential V(x) = Ax^2, where A is a positive real constant, will execute simple harmonic motion about the origin. Determine the period of the motion. Answer: An electron in a one-dimensional electric potential V(x) = Ax^2 will execute simple harmonic motion about the origin since the acceleration is proportional to -x with a constant of proportionality being 2A/m. The period of the motion is T = 2π√(m/2A).

Step-by-step solution

Calculating the force on the electron due to the electric potential

First, we need to find the force acting on the electron due to the electric potential, \(V(x) = Ax^2\). The force is given by the negative gradient of the potential, i.e., \(F(x) = - \dfrac{dV(x)}{dx}\). Now, let's calculate the force: $$ F(x) = -\frac{d}{dx}\left(Ax^2\right) = -2Ax $$

Determining the equation of motion

Since the force on the electron is given by \(F(x) = -2Ax\), we can formulate the equation of motion for the electron. The equation of motion is given by Newton's Second Law: \(F = ma\), where \(m\) is the mass of the electron and \(a\) is its acceleration. As the acceleration is the second derivative of the position with respect to time, we may write this as: $$ m\frac{d^{2}x}{dt^{2}} = -2Ax. $$

Identifying the properties of simple harmonic motion

We can see that the second derivative of \(x\) is proportional to \(-x\), specifically: $$ \frac{d^{2}x}{dt^{2}} = -\frac{2A}{m}x. $$ For simple harmonic motion, the acceleration of the object must be proportional to the displacement with a negative constant. In this case, we can see that the condition is fulfilled, since the acceleration is proportional to \(-x\) with the constant of proportionality being \(\frac{2A}{m}\). Therefore, we can conclude that the electron executes simple harmonic motion about the origin.

Finding the period of the motion

Now that we have confirmed that the motion is simple harmonic, we can determine the period of the motion, which is the time it takes for the electron to complete one full oscillation. The angular frequency, \(\omega\), of simple harmonic motion can be defined as: $$ \omega = \sqrt{\frac{k}{m}}, $$ where \(k = - \dfrac{d^2V}{dx^2}\) is the effective spring constant, and in our case is equal to \(2A\). Plugging the values into the equation, we get: $$ \omega = \sqrt{\frac{2A}{m}}. $$ The period, \(T\), of the motion is related to the angular frequency as follows: $$ T = \frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\dfrac{2A}{m}}}. $$ The period of the simple harmonic motion of the electron in the potential \(V(x) = Ax^2\) is given by: $$ T = 2\pi\sqrt{\frac{m}{2A}}. $$

Key concepts

Electrical Potential

Understanding the concept of electrical potential is fundamental in physics, especially when examining the motion of charged particles like electrons in fields. Electrical potential, often symbolized as 'V', represents the amount of work required to move a unit positive charge from a reference point to a specific point in the field without producing any acceleration. To visualize this, imagine carrying a rock uphill against gravity; the higher you go, the more work is required to combat gravity's pull. Similarly, in an electrical field, more work is needed to move a charge closer to another with the same polarity, as they repel each other.

The potential due to a one-dimensional electrical field like the one described by the equation \( V(x) = Ax^2 \), where \( A \) is a constant, shapes the path of the particle moving within it. When \( A \) is a positive real number, as stated in our example, the potential increases as the distance from the origin (\( x \)) increases. This creates a 'well' around the origin, guiding the particle to remain near it, which sets the stage for the possibility of simple harmonic motion (SHM) if the particle is displaced from the equilibrium position.

Equation of Motion

The equation of motion for a particle provides a mathematical relation between the position of a particle, its velocity, and its acceleration at any time. For our exercise concerning an electron in an electrical potential, we exploit Newton's second law, which states that the force exerted on an object is the product of its mass and its acceleration \( F = ma \).

To connect this with the electric potential, we recognize that the force experienced by the electron is the negative derivative of the potential energy function \( F(x) = - \frac{dV(x)}{dx} \). Simplifying the potential function given (\( V(x) = Ax^2 \)) through differentiation provides us the force, which we then equate to mass times acceleration. This yields the equation \( m\frac{d^{2}x}{dt^{2}} = -2Ax \), a second-order differential equation that defines the motion of our electron.

Insights into SHM

This particular form, where the second derivative of position (which is acceleration) is proportional to the position itself but negatively signed, is characteristic of simple harmonic motion – indicating that the electron will indeed oscillate about the origin, like a mass-spring system.

Period of Oscillation

The period of oscillation is a crucial aspect of simple harmonic motion, defining the time it takes for the system to complete one full cycle of motion. For any oscillating system, understanding the period helps in predicting the behavior of the oscillations. The period \( T \) is related to the angular frequency \( \omega \) by the relation \( T = \dfrac{2\pi}{\omega} \).

In our exercise, by equating the force to mass times acceleration, we were manipulating the equation of motion into a form similar to Hooke's law, which describes the restoring force in a spring-mass system. The result of step three in the solution gives the angular frequency \( \omega = \sqrt{\dfrac{2A}{m}} \). With that, we can find the period of the electron's motion. The square root in the angular frequency indicates that the period is inversely proportional to the square root of the constant \( A \), showing us how changing the value of \( A \) would affect the electron's oscillation cycle.

Therefore, to summarize, the period of oscillation for an electron in the given electric potential is expressed as \( T = 2\pi\sqrt{\frac{m}{2A}} \), which encapsulates the essence of its oscillatory motion: the greater the opposing force represented by \( A \), the faster the electron oscillates, thus the shorter the period, and vice versa.