Question

Obtain the Liénard-Wiechert potentials \((40.8)\) of an arbitrarily moving charge \(q\) by the following alternative method: Assume, first, that the charge moves uniformly and that in its rest frame the potential is given by \((40.2)\). Then transform this to the general frame, using the four-vector property of \(\Phi^{\mu}\). Finally extend the result to an arbitrarily moving charge by the argument we used after (40.2). [Hint: if the separation \((c t, r)\) between two events satisfies \(r=c t\) in one frame, it does so in all frames.]

Short answer

Question: Determine the Liénard-Wiechert potentials for an arbitrarily moving charge. Answer: The Liénard-Wiechert potentials for an arbitrarily moving charge can be found by following these steps: 1. Define the potential in the rest frame of the charge: $\Phi_0^{\mu} = (\Phi_0, \vec{0})$. 2. Transform the potential to the general frame using the Lorentz transformation: $\Phi^{\mu} = \Lambda^{\mu}_{\ \nu} \Phi_0^{\nu}$. 3. Determine the scalar potential $\Phi$ and the vector potential $\vec{A}$ in the general frame using the Lorentz transformation matrix. 4. Extend the result to an arbitrarily moving charge by replacing the constant velocity with the correct velocity in the expressions of $\Phi$ and $\vec{A}$, using the hint provided in the problem. The resulting expressions for the scalar potential $\Phi$ and the vector potential $\vec{A}$ will be the Liénard-Wiechert potentials for an arbitrarily moving charge.

Step-by-step solution

Define the potential in the rest frame of the charge

According to the problem, the potential in the rest frame of the uniformly moving charge is given by equation (40.2). Let's denote this potential as \(\Phi_0^{\mu} = (\Phi_0, \vec{0})\), where \(\Phi_0\) is the scalar potential in the rest frame and \(\vec{0}\) is the zero-vector since there is no vector potential in this frame.

Transform the potential to the general frame

To transform the potential to the general frame, we will use the four-vector property of \(\Phi^{\mu}\). Using the Lorentz transformation, we can find the components of the four-potential in the general frame. Let's denote the general frame potential as \(\Phi^{\mu} = (\Phi, \vec{A})\) and let the charge be moving with velocity \(\vec{v}\) in the general frame. The Lorentz transformation relates \(\Phi_0^{\mu}\) and \(\Phi^{\mu}\) as follows: $$ \Phi^{\mu} = \Lambda^{\mu}_{\ \nu} \Phi_0^{\nu} $$ where \(\Lambda^{\mu}_{\ \nu}\) is the Lorentz transformation matrix.

Determine the scalars and vectors of the potentials

Calculate the scalar potential \(\Phi\) and the vector potential \(\vec{A}\) in the general frame using the Lorentz transformation matrix from Step 2. Once you have these expressions, you have the Liénard-Wiechert potentials for the uniformly moving charge.

Extend the result for an arbitrarily moving charge

Now, we will extend the result from Step 3 to an arbitrarily moving charge. In order to do this, we need to employ the argument we used after equation (40.2). This involves using the hint provided in the problem (the separation between two events satisfies \(r = ct\) in one frame, then it holds true for all frames) and replacing the constant velocity with the correct velocity in the expressions of \(\Phi\) and \(\vec{A}\) that we just found. Once this substitution is made, the resulting expressions for \(\Phi\) and \(\vec{A}\) will be the Liénard-Wiechert potentials for an arbitrarily moving charge. These expressions will encompass the more general case and serve as the final solution to the problem.

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics developed by Albert Einstein, which revolutionized our understanding of space, time, and motion. It postulates that the laws of physics are the same for all non-accelerating observers and that the speed of light in a vacuum is the same for all observers, regardless of their relative motion.

Under special relativity, time and space are not absolute but are interwoven into a single continuum known as spacetime. Events that occur simultaneously for one observer may not be simultaneous for another, depending on their relative velocities. This leads to phenomena such as time dilation and length contraction, which means moving clocks tick slower and objects in motion shorten along the direction of travel. These counterintuitive effects are only noticeable at speeds approaching the speed of light.

In the exercise at hand, special relativity is crucial for transforming the electromagnetic potentials from the rest frame of a moving charge to a general frame of reference, taking into account that light's speed remains constant in all inertial frames.

Lorentz Transformation

The Lorentz transformation is a set of equations in special relativity that describes how, according to different observers, the time and space coordinates of events change when transitioning from one inertial frame to another. These transformations ensure that the laws of physics, including the invariant speed of light, hold true in all inertial frames.

Mathematically, the Lorentz transformation relates the space and time coordinates \( (t, x, y, z) \) in one inertial frame to the coordinates \( (t', x', y', z') \) in another, moving at a constant velocity relative to the first. The transformation is governed by parameters like the relative velocity between the observer and the moving frame, and the Lorentz factor, which accounts for time dilation and length contraction.

The exercise solution involves using the Lorentz transformation to modify the potentials associated with a uniformly moving charge to those perceived in another inertial frame. This technique is necessary to calculate the Liénard-Wiechert potentials for an arbitrarily moving charge.

Four-Vector

A four-vector is an object in the realm of special relativity that combines both spatial and temporal components into a single entity. In a four-dimensional spacetime, these vectors consist of a time component typically multiplied by the speed of light to ensure that all components have the same dimension and three spatial components. Examples of four-vectors include the position four-vector, velocity four-vector, and the four-momentum.

The transformation properties of four-vectors are particularly important because they ensure that physical laws are invariant under Lorentz transformations, meaning they hold true in all inertial frames. This is why four-vectors are fundamental to modern physics calculations.

In our exercise, the electromagnetic potential \( \Phi^{\mu} = (\Phi, \vec{A}) \) is treated as a four-vector. This is crucial when transforming the potentials from the rest frame of the charge to those observed in another frame, ensuring that the transformation adheres to the principles of special relativity.

Electromagnetic Potential

Electromagnetic potential, comprising the scalar potential \( \Phi \) and the vector potential \( \vec{A} \), is a means of expressing the electromagnetic field. These potentials are incredibly useful in the study of electrodynamics because they offer a simplified way to calculate electric and magnetic fields. The scalar potential \( \Phi \) relates to the electric field, while the vector potential \( \vec{A} \) is associated with the magnetic field.

The beauty of working with these potentials is that they obey the principle of gauge invariance, meaning different potentials can result in the same electromagnetic field. This flexibility can significantly simplify calculations.

In the context of the exercise, by employing special relativity and the properties of four-vectors, one derives the Liénard-Wiechert potentials. These potentials describe the fields due to a moving charge and are essential for understanding electromagnetic phenomena involving relativistic speeds, which is precisely what the problem requires us to determine.