🔑:## Step 1: Define the energy of interaction between a magnetic moment and a magnetic field.The energy of interaction (E) between a magnetic moment μ and a magnetic field B is given by the scalar product of μ and B: E = -μ∙B.## Step 2: Express the magnetic moment μ in terms of the orbital angular momentum l.For an electron, the magnetic moment μ due to its orbital motion can be expressed as μ = (e/2m)l, where e is the charge of the electron, m is its mass, and l is the orbital angular momentum.## Step 3: Express the magnetic field B in terms of the orbital angular momentum l.The magnetic field B generated by the orbital motion of the electron can be expressed using the Biot-Savart law. However, for the purpose of this problem, we consider B to be proportional to l, since the magnetic field generated by a current loop (which is analogous to the electron's orbit) is proportional to the current, and the current is proportional to the velocity of the electron, which in turn is related to its orbital angular momentum.## Step 4: Introduce the spin angular momentum s and its relation to the magnetic moment.The spin angular momentum s of the electron also generates a magnetic moment, given by μ_s = (e/2m)s for the spin contribution. However, the problem focuses on the orbital contribution to the magnetic field and its interaction with the spin.## Step 5: Derive the energy of interaction in terms of s and l.Substituting μ = (e/2m)l into the energy equation E = -μ∙B and recognizing that B is proportional to l (from the orbital motion), we get E ∝ -l∙(l). However, the question asks for the interaction in terms of s∙l, suggesting we consider how the spin magnetic moment interacts with the field generated by the orbital motion. Thus, considering μ_s = (e/2m)s interacting with B ∝ l, the energy of interaction E ∝ -s∙l, because the magnetic field generated by the orbital angular momentum (l) interacts with the magnetic moment due to the spin (s).## Step 6: Explain the physical significance of the result.The result indicates that the energy of interaction between the spin magnetic moment and the magnetic field generated by the orbital motion of the electron is proportional to the scalar product of the spin and orbital angular momenta (s∙l). This interaction is known as the spin-orbit interaction and is responsible for the fine structure in atomic spectra. It implies that the energy levels of an atom depend not only on the orbital angular momentum but also on the relative orientation of the spin and orbital angular momenta.The final answer is: boxed{E ∝ -s∙l}

🔑:In Jane Austen's Persuasion, Anne Elliot undergoes significant character development, particularly in relation to her interactions with Captain Wentworth and her family members. The events of Chapters 19-24 are pivotal in Anne's growth and self-awareness, as she navigates her feelings, confronts her past mistakes, and reevaluates her relationships.Anne's interactions with Captain Wentworth:In Chapters 19-24, Anne's interactions with Captain Wentworth reveal her deep-seated emotions and her gradual recognition of her own desires. Initially, Anne is hesitant to express her feelings, fearing rejection and humiliation. However, as she spends more time with Captain Wentworth, she begins to confront her past mistakes and the pain she has endured. The awkward encounters between Anne and Captain Wentworth, particularly in Chapter 20, demonstrate Anne's lingering emotions and her struggle to reconcile her past and present feelings.As the chapters progress, Anne's interactions with Captain Wentworth become more open and honest. In Chapter 23, Captain Wentworth's letter, which reveals his enduring love for Anne, marks a turning point in their relationship. Anne's response, in which she acknowledges her own feelings and regrets, demonstrates her growing self-awareness and emotional maturity. This exchange highlights the theme of love as a redemptive force, capable of overcoming past mistakes and social obstacles.Anne's interactions with her family members:Anne's relationships with her family members, particularly her father and sister Mary, also contribute to her character development. In Chapters 19-24, Anne's interactions with her family reveal her growing independence and self-respect. She begins to challenge her family's values and expectations, particularly in regards to her relationship with Captain Wentworth. Anne's conversation with her father in Chapter 22, in which she defends her decision to accept Captain Wentworth's proposal, demonstrates her increased confidence and assertiveness.Anne's interactions with her sister Mary, who embodies the selfish and snobbish aspects of the Elliot family, serve as a foil to Anne's growth. Mary's constant complaints and manipulations highlight the flaws of the aristocratic class, while Anne's patience and kindness demonstrate her own moral integrity. Through these interactions, Austen critiques the social conventions that prioritize family connections and material status over personal happiness and moral character.Character development and themes:The events of Chapters 19-24 contribute significantly to Anne's growth and self-awareness, as she:1. Confronts her past mistakes: Anne acknowledges her error in rejecting Captain Wentworth's proposal and begins to make amends.2. Develops emotional maturity: Anne learns to express her feelings and desires, rather than suppressing them to conform to societal expectations.3. Asserts her independence: Anne challenges her family's values and expectations, demonstrating her growing self-respect and confidence.4. Reevaluates her relationships: Anne reassesses her connections with her family members and Captain Wentworth, prioritizing personal happiness and moral character over social status.These developments reveal the novel's themes of:1. Love as a redemptive force: The novel highlights the power of love to overcome past mistakes and social obstacles, as embodied in the reunion of Anne and Captain Wentworth.2. Class and social status: Austen critiques the social conventions that prioritize family connections and material status over personal happiness and moral character, as exemplified by the Elliot family's snobbery and selfishness.3. Personal growth and self-awareness: The novel emphasizes the importance of self-reflection, emotional maturity, and independence in achieving personal happiness and fulfillment, as demonstrated by Anne's character development.In conclusion, the events of Chapters 19-24 in Persuasion are crucial to Anne Elliot's character development, as she navigates her feelings, confronts her past mistakes, and reevaluates her relationships. Through Anne's growth and self-awareness, Austen explores the novel's themes of love, class, and social status, ultimately highlighting the importance of personal happiness, moral character, and emotional maturity in achieving fulfillment.

🔑:The annihilation of matter and antimatter is a fundamental process in physics that arises from the interactions between particles and antiparticles, which are described by the Standard Model of particle physics. The key to understanding this process lies in the interaction terms of the Lagrangian and the concept of quantum fields.## Step 1: Understanding the Lagrangian and Interaction TermsThe Lagrangian is a mathematical function that describes the dynamics of a physical system. In the context of particle physics, the Lagrangian includes terms that represent the kinetic energy of particles, their mass, and the interactions between different particles. For matter and antimatter, the interaction terms in the Lagrangian involve the exchange of gauge bosons (like photons for electromagnetic interactions, gluons for strong interactions, and W and Z bosons for weak interactions).## Step 2: Quantum Fields and Particle-Antiparticle CreationIn quantum field theory, particles and antiparticles are viewed as excitations of underlying quantum fields. Each type of particle (electron, quark, etc.) has its corresponding antiparticle (positron, antiquark, etc.), and the quantum field associated with a particle can create or annihilate both the particle and its antiparticle. This creation and annihilation process is fundamental to understanding how matter and antimatter interact.## Step 3: Annihilation ProcessWhen a particle meets its antiparticle, they can annihilate each other. This process is facilitated by the interaction terms in the Lagrangian, which allow for the exchange of particles that mediate the force between the particle and antiparticle. For example, an electron and a positron can annihilate into two photons (the particles that mediate the electromagnetic force), with the energy and momentum of the electron-positron pair being converted into the energy and momentum of the photons.## Step 4: Conservation Laws and SymmetriesThe annihilation process must obey various conservation laws, such as conservation of energy, momentum, charge, and spin. These laws, along with the symmetries of the Standard Model (like charge conjugation, parity, and time reversal symmetry), dictate the possible outcomes of particle-antiparticle interactions, including annihilation.## Step 5: Fundamental Physical ReasonThe fundamental physical reason behind the annihilation of matter and antimatter is rooted in the structure of the quantum fields and the interaction terms in the Lagrangian. The ability of quantum fields to create and annihilate particles and antiparticles, combined with the specific interactions mediated by gauge bosons, leads to the annihilation process. This process is a direct consequence of the underlying symmetries and conservation laws that govern the behavior of particles and fields in the universe.The final answer is: boxed{The interaction terms in the Lagrangian and the concept of quantum fields, which allow for the creation and annihilation of particles and antiparticles, are the fundamental reasons behind the annihilation of matter and antimatter.}

🔑:## Step 1: Determine the relationship between the electric and magnetic fields.The equation Emax = Bmax*c shows that the maximum electric field (Emax) is equal to the maximum magnetic field (Bmax) times the speed of light (c). Since we're given the rms (root mean square) strength of the magnetic field, we first need to find the maximum magnetic field. The relationship between rms and maximum values for a sinusoidal wave is Bmax = sqrt(2)*Brms.## Step 2: Calculate the maximum magnetic field (Bmax).Given Brms = 7.8 * 10^-9 T, we can calculate Bmax = sqrt(2)*Brms = sqrt(2)*7.8 * 10^-9 T.## Step 3: Calculate Bmax value.Bmax = sqrt(2)*7.8 * 10^-9 T = 1.1 * 10^-8 T (approximately, since sqrt(2) is approximately 1.414).## Step 4: Calculate the maximum electric field (Emax).Using the equation Emax = Bmax*c, where c = 3 * 10^8 m/s, we can find Emax = 1.1 * 10^-8 T * 3 * 10^8 m/s.## Step 5: Perform the calculation for Emax.Emax = 1.1 * 10^-8 T * 3 * 10^8 m/s = 3.3 * 10^0 N/C = 3.3 N/C.## Step 6: Determine the rms electric field (Erms).The rms electric field can be found from the maximum electric field using the relationship Erms = Emax/sqrt(2).## Step 7: Calculate Erms.Erms = 3.3 N/C / sqrt(2) = 3.3 N/C / 1.414 = 2.33 N/C.## Step 8: Determine the frequency of the electric field.The frequency of the electric field is the same as the frequency of the magnetic field since they are part of the same electromagnetic wave. Thus, the frequency of the electric field is 88.8 kHz.## Step 9: Determine the direction of the electric field.Since the magnetic field is oscillating vertically and the wave is traveling west, the electric field must be oscillating in a direction perpendicular to both the magnetic field and the direction of propagation. By the right-hand rule, if you point your thumb west (direction of propagation) and your fingers up (direction of B), your palm will face south, indicating the direction of the electric field is south-north.The final answer is: boxed{2.33}