Question

A point charge \(+q\) rests halfway between two steady streams of positive charge of equal charge per unit length \(\lambda\), moving opposite directions and each at \(c / 3\) relative to point charge. With equal electric forces on the point charge, it would remain at rest. Consider the situation from a frame moving right at \(c / 3\). (a) Find the charge per unit length of each stream in this frame. (b) Calculate the electric force and the magnetic force on the point charge in this frame, and explain why they must be related the way they are. (Recall that the electric field of a line of charge is \(\lambda / 2 \pi \varepsilon_{0} r\), that the magnetic field of a long wire is \(\mu_{0} I / 2 \pi r\), and that the magnetic force is \(q \mathbf{v} \times \mathbf{B}\). You will also need to relate \(\lambda\) and the current \(L\).)

Short answer

The charge per unit length of each stream in the moving frame is given by the Lorentz-transformed value of the charge per unit length in the rest frame. The electric and magnetic forces on the point charge in this frame can be calculated using standard formulas for electric force (\(F_e' = q E'\)) and magnetic force (\(F_m' = qv B'\)), with the electric and magnetic fields given by the appropriate expressions for a line charge and a current-carrying wire, respectively. The electric and magnetic forces must balance out for the point charge to remain at rest in this moving frame.

Step-by-step solution

Apply Relativistic Transformation to Find Charge Density from a Moving Frame

Considering the situation from a frame moving right at \(c/3\), due to relativistic effects, the charge per unit length in the frame moving with velocity \(c/3\) will be given by \(\lambda' = \gamma \lambda\) where \(\gamma = 1 / \sqrt{1 - (v/c)^2}\) is the Lorentz factor. Here \(v = c/3\) and \(\lambda\) is the charge per unit length as observed from the rest frame. Plug in the values to get \(\lambda'\).

Calculate Electric Force in the Moving Frame

In the moving frame, the electric field due to each line charge at the location of the point charge is given by \(E' = \lambda' / 2 \pi \varepsilon_{0} r\). The total electric field at the location of the point charge is twice that value (due to two streams), so the electric force on the point charge is \(F_e' = 2 q E' = q \lambda' / \pi \varepsilon_{0} r\).

Relate Charge Density and Current

The current in each wire is \(I = \lambda v = \lambda (c/3)\) as each stream moves at \(c/3\) relative to the stationary point charge. With the same velocity in the moving frame, the current in each wire in the moving frame is \(I' = \lambda' v\).

Calculate Magnetic Force in the Moving Frame

The magnetic field due to each wire at the location of the point charge is \(B' = \mu_{0} I' / 2 \pi r\). The magnetic force on the point charge is \(F_m' = qvB' = qv \mu_{0} I' / 2 \pi r = qv \mu_{0} \lambda' v / 2 \pi r\).

Establish Relationship between Electric and Magnetic Forces

Note that the electric force and magnetic force must balance out for the point charge to remain at rest in this moving frame. Therefore, the electric force obtained in Step 2 should equal the magnetic force obtained in Step 4. This provides the necessary relation between electric and magnetic forces.

Key concepts

Lorentz Factor

The Lorentz factor is a crucial quantity in special relativity, which comes into play when analyzing situations involving objects moving at significant fractions of the speed of light. It is represented by the Greek letter \( \gamma \) and is defined as \( \gamma = \frac{1}{\sqrt{1 - (v/c)^2}} \), where \( v \) is the velocity of the moving object and \( c \) is the speed of light. As the velocity \( v \) approaches the speed of light, \( \gamma \) increases dramatically, which reflects the substantial differences between relativistic and classical mechanics predictions.

The Lorentz factor is essential for understanding how time, length, and mass change with velocity. For instance, it predicts that time will appear to move slower (time dilation) and lengths will contract (length contraction) in the frame that is moving relative to an observer. In the exercise provided, the charge density observed in a moving frame is amplified by the Lorentz factor because, from that frame's perspective, the distances between charges contract. This is why the charge per unit length \( \lambda' \) in the moving frame, where \( v = c/3 \) is given by \( \lambda' = \gamma \lambda \). The Lorentz factor plays an instrumental role in all calculations involving relativity, ensuring that the laws of physics hold true in all inertial frames.

Electric Force in a Moving Frame

In classical electromagnetism, the electric force felt by a charge is usually determined by the electric field in the same reference frame. However, when the frame is moving, relativistic effects must be considered. In a moving frame, the charge density changes due to length contraction, which in turn alters the perceived electric field.

As we examine the electric force on a charge in a moving frame, it is altered by the transformed charge density. To find this force, we first calculate the altered electric field \( E' \) using the transformed charge per unit length \( \lambda' \) and then multiply by the charge \( q \) to get the force \( F_e' \). This way, relativity directly impacts the electric force calculations for charges in relative motion. The exercise demonstrates this by considering the electric force in a frame moving with the charge streams. The result shows that the concept of a static electric field does not hold in every frame but must be adjusted for the effects of relativity.

Magnetic Force on a Moving Charge

A magnetic force arises from a moving charge's interaction with a magnetic field. According to the Lorentz force law, a charge \( q \) moving with velocity \( \mathbf{v} \) in a magnetic field \( \mathbf{B} \) experiences a force \( \mathbf{F_m} = q \mathbf{v} \times \mathbf{B} \). This cross product ensures that the direction of the magnetic force is perpendicular to both \( \mathbf{v} \) and \( \mathbf{B} \).

In the context of the provided exercise, this concept shows that a stationary point charge in one frame may experience a magnetic force in another frame if there is relative motion between the charge and the magnetic field source. To determine the magnetic force in the moving frame, the current, which is related to the charge density, must first be calculated. Then, using the Biot-Savart law, the magnetic field \( \mathbf{B'} \) due to the current is found. Finally, plugging this into the Lorentz force equation gives the resultant magnetic force \( F_m' \) on the point charge. This illustrates the deep connection in electromagnetism between electricity and magnetism, particularly when charges are in motion.

Relativity in Electromagnetism

The theory of relativity has profound implications in electromagnetism, specifically in how electric and magnetic fields transform between different inertial frames. According to special relativity, observers moving at constant velocities (inertial frames) may perceive electric and magnetic fields differently. The electric field in one frame can be seen as a combination of electric and magnetic fields in another frame. This shows that electric and magnetic forces are not independent but rather intimately connected through the principles of relativity.

In practice, this means that a purely electric force in one reference frame could be perceived as a mix of electric and magnetic forces in another. The problem set illustrates this by showing that what is balanced electric force in one frame must be countered by electric and magnetic forces in another frame moving relative to the first. Consequently, special relativity ensures that the principles of electromagnetism apply consistently across different reference frames and that the laws of physics are the same for all observers in uniform motion, forming the cornerstone of modern physics.