Question

A proton enters the gap between a pair of metal plates (an electrostatic separator) that produces a uniform, vertical electric field between them. Ignore the effect of gravity on the proton. a) Assuming that the length of the plates is \(15.0 \mathrm{~cm}\), and that the proton will approach the plates at a speed of \(15.0 \mathrm{~km} / \mathrm{s}\) what electric field strength should the plates be designed to provide, if the proton must be deflected vertically by \(1.50 \cdot 10^{-3} \mathrm{rad} ?\) b) What speed does the proton have after exiting the electric field? c) Suppose the proton is one in a beam of protons that has been contaminated with positively charged kaons, particles whose mass is \(494 \mathrm{MeV} / \mathrm{c}^{2}\left(8.81 \cdot 10^{-28} \mathrm{~kg}\right)\), compared to the mass of the proton, which is \(938 \mathrm{MeV} / \mathrm{c}^{2}\left(1.67 \cdot 10^{-27} \mathrm{~kg}\right)\) The kaons have \(+1 e\) charge, just like the protons. If the electrostatic separator is designed to give the protons a deflection of \(1.20 \cdot 10^{-3} \mathrm{rad}\), what deflection will kaons with the same momentum as the protons experience?

Short answer

Answer: The required electric field strength is approximately \(7.49 \times 10^6\) N/C, the final speed of the proton remains the same at 15.0 km/s, and the deflection experienced by the charged kaons is approximately \(6.32 \times 10^{-4}\) rad.

Step-by-step solution

Part A - Electric field strength

To find the required electric field strength, we can use the deflection formula for charged particles passing through an electric field: \(\theta = \frac{eE \ell}{m v^2}\), where \(\theta\) is the deflection angle, \(e\) is the charge of the particle, \(E\) is the electric field strength, \(\ell\) is the length of the plates, \(m\) is the mass of the particle, and \(v\) is the initial speed of the particle. We can rearrange the formula to solve for \(E\) and plug in the given values. \(E = \frac{\theta m v^2}{e\ell}\) Given: \(\theta = 1.50 \cdot 10^{-3} \mathrm{rad}\), \(\ell = 15.0 \times 10^{-2} \mathrm{m}\), \(v = 15.0 \times 10^3 \mathrm{m/s}\), \(m = 1.67 \times 10^{-27} \mathrm{kg}\), and \(e = 1.60 \times 10^{-19} \mathrm{C}\). \(E = \frac{(1.50 \cdot 10^{-3})(1.67 \times 10^{-27})(15.0 \times 10^3)^2}{(1.60 \times 10^{-19})(15.0 \times 10^{-2})}\) \(E \approx 7.49 \times 10^6 \mathrm{N/C}\) So, the needed electric field strength is approximately \(7.49 \times 10^6 \mathrm{N/C}\).

Part B - Final speed of proton

The initial kinetic energy of the proton is \(K_i = \frac{1}{2} m v^2\). After passing through the electric field, the proton has lost some potential energy, but its vertical velocity remains unchanged (since horizontal motion is independent from vertical motion). Thus, the final kinetic energy is still \(K_f = \frac{1}{2} m v^2\). Therefore, the final speed of the proton remains the same at \(15.0 \mathrm{~km/s}\).

Part C - Deflection of charged kaons

To find the deflection experienced by the charged kaons with the same momentum as the protons, we will use the same deflection formula as before: \(\theta = \frac{eE \ell}{m v^2}\). Since the momentum is the same, \(m v\) is the same for both protons and kaons, but the mass is different. Given that the mass of a kaon is \(m_k = 8.81 \times 10^{-28}\) kg, the mass ratio between a kaon and a proton is: \(r = \frac{m_k}{m_p} = \frac{8.81 \times 10^{-28}}{1.67 \times 10^{-27}} \approx 0.527\) Now we can find the deflection angle for the kaons, \(\theta_k\), by multiplying the deflection angle for the protons, \(\theta_p = 1.20 \times 10^{-3} \mathrm{rad}\), by the mass ratio: \(\theta_k = r \times \theta_p \approx 0.527 \times 1.20 \times 10^{-3} = 6.32 \times 10^{-4} \mathrm{rad}\) The deflection experienced by the kaons is approximately \(6.32 \times 10^{-4} \mathrm{rad}\).

Key concepts

Charged Particle Trajectories

When a charged particle, such as a proton or an electron, enters an electric field, it experiences a force due to its charge. This force alters the particle's path, resulting in a trajectory that can be predicted and calculated.

For instance, if a proton enters a region with a uniform electric field, it will begin to move in a parabolic path if it is initially travelling perpendicular to the field lines. If it enters at an angle, the trajectory becomes more complex due to the combination of initial velocity components and the acceleration caused by the electric field.

The degree to which the particle is deflected depends on several factors, including the strength of the electric field, the length of the field region, the speed of the particle, and the particle's charge and mass. By understanding these relationships, we can design equipment, such as the electrostatic separator in our exercise, to manipulate particle trajectories for various applications in scientific research and industry.

Uniform Electric Field

A uniform electric field is characterized by electric field lines that are parallel and equally spaced, indicating that the force experienced by a charged particle at any point within the field is constant both in magnitude and direction.

Uniform electric fields are often created between two parallel metal plates with a voltage difference applied across them. In our exercise, the metal plates are used to deflect the path of a proton, which is a common scenario in physics experiments.

Knowing the strength of the electric field (measured in Newtons per Coulomb, N/C) allows us to determine how much force a charged particle will experience and to calculate the resulting deflection of its trajectory when it passes between the plates.

Particle Physics

Particle physics is a field of study that delves into the nature of particles that are the fundamental constituents of matter and radiation. It endeavors to understand the forces that govern their interactions and behavior.

In the context of our problem, both protons and kaons are subatomic particles that are subject to the fundamental forces of nature, including the electromagnetic force. A clear understanding of these forces and particle properties, such as mass and charge, is essential for predicting how particles will behave when subjected to electric fields.

Charged particle beams, which are streams of particles like protons or kaons, are used in accelerators for research purposes, including probing the internal structure of particles or colliding particles at high energies to discover new phenomena.

Momentum Conservation

The principle of momentum conservation states that in a closed system with no external forces, the total momentum remains constant over time. This principle is central to many phenomena in physics and is heavily utilized in particle physics.

In our exercise scenario, when considering particles like protons and kaons with the same momentum, we're working under the assumption that the product of mass and velocity (momentum) for each particle type is identical as they enter the electric field. The differences in deflection observed between protons and kaons do not violate conservation of momentum because it is the electromagnetic force exerted by the field that's responsible for changing their trajectories.

The uniform electric field does not alter the horizontal component of the particles' momentum, thus ensuring that momentum in that direction is conserved. By controlling the electric field, we can separate particles with different masses but the same charge, providing a clear demonstration of momentum conservation at work.