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Acquiring transverse momentum In the rest frame of a particle with charge \(q_{1}\), another particle with charge \(q_{2}\) is approaching, moving with velocity \(v\) not small compared with \(c\). If it continues to move in a straight line, it will pass a distance \(b\) from the position of the first particle. It is so massive that its displacement from the straight path during the encounter is small compared with \(b\). Likewise, the first particle is so massive that its displacement from its initial position while the other particle is nearby is also small compared with \(b\).(a) Show that the increment in momentum acquired by each particle as a result of the encounter is perpendicular to \(\mathbf{v}\) and has magnitude \(q_{1} q_{2} / 2 \pi \epsilon_{0} v b\). (Gauss's law can be useful here.) (b) Expressed in terms of the other quantities, how large (in order of magnitude) must the masses of the particles be to justify our assumptions?

Step by step solution

01

Express the Electric Field

The electric field E produced by the charge \( q_{1} \) can be given by Gauss's Law as \( E = q_{1} / 4 \pi \epsilon_{0} r^{2} \) where r is the distance from the charge.

02

Compute the Increment in Momentum

Charges in an electric field will experience a change in momentum. In this scenario, this change is given by \( \Delta p = \int_{-\infty}^{\infty} q_{2} E dt \). After integrating, the increment in momentum results in \( \Delta p = q_{1} q_{2} / 2 \pi \epsilon_{0} v b \) which is perpendicular to \( \mathbf{v} \). This shows that the increment in momentum acquired by each particle as a result of the encounter is perpendicular to \( \mathbf{v} \).

03

Derive the Condition for the Masses

The presence of the other particle should not significantly influence the motion of either particle. This means the displacement must be less than \( b \). This is achieved if the acceleration \( a = F/m \) during the encounter is much less than \( v^{2} / b \). Using \( F = q_{1} q_{2} / 4 \pi \epsilon_{0} b^{2} \) and substituting for the acceleration, we get \( m \gg q_{1} q_{2} / 4 \pi \epsilon_{0} v^{2} b \). This justifies our initial assumptions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Fields

Electric fields are invisible forces surrounding charged particles. They exert a force on other charged objects within the field. The intensity of this force depends on both the charge creating the field and the distance from the charge.

Electric fields are calculated using Coulomb's law, which states that the electric field (E) due to a point charge is given by \[ E = \frac{q}{4 \pi \epsilon_0 r^2} \]where:

• \( q \) is the charge of the particle.
• \( \epsilon_0 \) is the permittivity of free space.
• \( r \) is the distance from the charge.

These fields influence how charges interact as they move through each other's fields, causing forces that can alter their motion. This concept is key to understanding how charges acquire momentum when passing near each other.

Gauss's Law

Gauss's Law is important in understanding electric fields. It relates the electric flux flowing out of a closed surface to the charge enclosed within that surface.

The mathematical form of Gauss's Law is:\[ \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]where:

• \( \mathbf{E} \) is the electric field.
• \( d\mathbf{A} \) is the differential area on the closed surface.
• \( Q_{\text{enc}} \) is the total charge enclosed inside the surface.

This law is especially useful because it allows us to calculate electric fields of symmetrical charge distributions much easier than using Coulomb's law directly. In the original problem, it helps determine the field influencing the moving charge, especially when distances and symmetrical considerations are involved, like maintaining the straight-line path assumption between massive particles.

Particle Physics

Particle physics explores the fundamental particles and forces, impacting how charged particles interact. It involves both the electromagnetic forces (as seen with electric fields and Gauss's law) and weak, strong, and gravitational forces in other contexts.

For the described scenario of two charged particles, the focus is on how the electromagnetic forces influence their trajectories and momentum. A particle's charge and mass are vital characteristics that dictate its behavior in a field. High energy exchanges are typical, requiring considerations of mass and energy, as motion can be nearly at the speed of light. Ideas from particle physics allow us to predict particle behavior when they are significantly influenced by short-range and long-range forces.

Relativistic Velocities

Relativistic velocities occur when particles move at speeds close to the speed of light (denoted \( c \)). At these speeds, Newtonian mechanics no longer apply, and special relativity principles come into play.

Key effects of relativistic velocities include:

• Time dilation, where time appears to slow down relative to an observer.
• Length contraction, where lengths parallel to the direction of motion shorten.
• Relativistic mass, where the mass of the object increases with velocity.

In the context of momentum transfer between charged particles, relativistic effects dictate how mass and energy should be factored into calculations, helping ensure predictions and observations match. Accurate assessment of how fast-moving charges exchange momentum often relies on relativistic considerations, particularly when velocities are not small compared to \( c \).