Question

In an inertial frame there is a uniform electric field e in the positive \(x\)-direction, which exerts a rest mass preserving force \(\mathbf{f}=q \mathbf{e}\) on a particle carrying a charge \(q\). If the particle is released from rest at the origin, and \(m, \tau\) denote its rest mass and proper time, prove that the particle will move (hyperbolically) according to the equation

Short answer

Question: Prove that the motion of a charged particle in an inertial frame with a uniform electric field in the x-direction follows a hyperbolic trajectory. Answer: By analyzing the force acting on the charged particle due to the electric field and applying Newton's second law, we derived the equation for the x-component of motion as \(x(\tau) = \frac{qe\tau^2}{2m}\). Since the x-component changes quadratically with respect to proper time τ and the y and z components remain constant at 0, the motion of the charged particle follows a hyperbolic trajectory in the x-direction.

Step-by-step solution

1. Analyzing the Force and Equations of Motion

For a charged particle with charge q in an electric field \(\mathbf{e}\), the force acting on the particle \(\mathbf{f}\) is given by: \(\mathbf{f} = q\mathbf{e}\) Since the particle is in an inertial frame, we can use Newton's second law to write the equation of motion: \(m\frac{d^2\mathbf{x}}{d\tau^2} = \mathbf{f}\) As the electric field is only in the positive x-direction, the force will only have an x-component: \(\mathbf{f} = (q\mathbf{e}, 0, 0) = (qe, 0, 0)\) This means that the particle is only experiencing a force in the x-direction.

2. Solving for the x-component of Motion

Now, let's analyze the x-component of the equation of motion: \(m\frac{d^2x}{d\tau^2} = qe\) To solve for x as a function of proper time τ, we can integrate this equation twice with respect to τ: \(\frac{dx}{d\tau} = v_x = \int \frac{qe}{m}d\tau\) \(v_x = \frac{qet}{m} + C_1\) Next, we'll integrate again with respect to τ: \(x = \int v_x d\tau = \int \left(\frac{qet}{m} + C_1\right)d\tau\) \(x = \frac{qe\tau^2}{2m} + C_1\tau + C_2\) Since the particle is released from rest at the origin, we know that at τ=0, x=0, and \(v_x=0\). We can use these initial conditions to find the values of the constants C_1 and C_2. - At τ=0, x=0: $ 0 = \frac{qe(0)^2}{2m} + C_1(0) + C_2 \Rightarrow C_2=0 $ - At τ=0, \(v_x=0\): $ 0 = \frac{qe(0)}{m} + C_1 \Rightarrow C_1=0 $ With these constants found, we can state the equation for the x-component of motion: \(x(\tau) = \frac{qe\tau^2}{2m}\)

3. Formulating the Hyperbolic Motion Equation

Now that we have derived the equation for the x-component of motion, we can write the full motion equation as follows: \(\mathbf{x}(\tau) = \left(\frac{qe\tau^2}{2m}, 0, 0\right)\) This equation represents a hyperbolic motion in the x-direction, as the position x changes quadratically with respect to proper time τ, while the y and z components remain constant at 0. Thus, we have proven that the particle's motion in the inertial frame with a uniform electric field in the x-direction is hyperbolic.

Key concepts

Hyperbolic Motion

Hyperbolic motion in physics refers to the trajectory of an object when it experiences constant acceleration. This type of motion is described using hyperbolas, which are open curves similar to parabolas. In a uniform electric field, a charged particle will experience this kind of motion due to the constant force applied to it.

When a particle is subjected to a steady force, as is the case with a charged particle in an electric field, the acceleration remains unchanged. This constant acceleration results in a motion of the particle where its velocity changes linearly with time, and its position changes according to the square of the time elapsed. Interestingly, this relationship closely resembles the equations that define a hyperbola in Cartesian coordinates.

Considering our exercise, the charged particle moves along a trajectory that can be plotted as a hyperbolic path on a graph with position on one axis and time on another. This comes from integrating the force felt by the particle, which is a product of the charge and the electric field strength. Hyperbolic motion is significant in physics as it helps in understanding the behavior of particles under uniform forces and has implications in fields such as electromagnetism and special relativity.

Proper Time

Proper time, denoted as \(\tau\), is a concept from Einstein's theory of special relativity that represents the time as measured by a clock moving with the object. It is the time interval between two events that occur in the same place in the reference frame of the moving object or observer.

In the context of our exercise, proper time is used to describe the motion of the charged particle. It is the time that ticks by from the perspective of the particle itself. This is crucial since the rate of time can differ for observers in relative motion due to the effects of time dilation. However, for an object moving in a uniform electric field, the proper time provides us a consistent way to measure the progression of phenomena, allowing us to integrate the forces and accelerations experienced by the particle over its own timeframe.

Understanding proper time is essential for solving problems in relativistic physics where observers may be in different frames of reference. It is the 'personal' time for the particle, undisturbed by the relative motion seen by external observers, and it offers a ground for accurate physical predictions in the particle's frame.

Rest Mass

Rest mass, often denoted by \(m\), is the mass of a physical object when it is at rest in an inertial frame of reference. It's a fundamental property that remains constant regardless of the object's velocity or the observer's frame of reference — a postulate sustained by the theory of special relativity.

In our exercise scenario, the rest mass represents the intrinsic mass of the charged particle that is not affected by its motion, allowing us to predict its dynamic behavior under the influence of forces. When the term \(m\) is used in equations of motion, as it is in our exercise's step-by-step solution, it embodies a measure of the particle's inertia or resistance to changes in its state of motion.

Because the rest mass is an invariant quantity, it plays a pivotal role in the laws of physics, including the principle underlying the famous mass-energy equivalence \(E=mc^2\), which dictates that mass can be converted into energy (and vice versa). Rest mass is also key in calculating gravitational attraction, affecting how objects move in a gravitational field.

Uniform Electric Field

A uniform electric field is characterized by having the same magnitude and direction at every point in space. It is often visualized by a series of parallel and equally spaced field lines. In such a field, the force experienced by a charged particle is constant both in magnitude and direction, leading to the aforementioned hyperbolic motion when the particle's initial velocity is zero.

In the problem at hand, the electric field is described as uniform and in the positive x-direction, implying that the entire space considered has a consistent, unchanging influence on the charged particle. The consistency of the electric field is what makes the force on the particle constant, allowing us to derive simple equations of motion and predict the particle's trajectory with precision.

The concept of a uniform electric field is a great simplification in many physics problems, including those related to capacitors, where it is a good approximation for the field between two closely spaced, parallel conducting plates. This simplification significantly eases calculations and is an essential element in understanding fundamental electric forces and their impact on matter.