🔑:## Step 1: Understanding the ProblemThe problem involves a negatively charged body, such as an electret, placed in a container of NaCl solution. We need to describe the behavior of the positive sodium ions in relation to the charged body, considering the electrostatic potential and the role of mobile ions in the solution.## Step 2: Electrostatic Potential and Mobile IonsWhen a negatively charged body is introduced into an electrolyte solution like NaCl, it creates an electrostatic potential that influences the distribution of ions in the solution. The positive sodium ions (Na+) are attracted to the negatively charged body due to electrostatic forces, while the negative chloride ions (Cl-) are repelled.## Step 3: Poisson-Boltzmann EquationThe Poisson-Boltzmann equation is a fundamental equation in electrostatics that relates the electrostatic potential to the distribution of charges in a system. For a solution containing mobile ions, it can be written as:[nabla^2 psi = -frac{rho}{epsilon} = -frac{1}{epsilon} sum_i z_i e c_i expleft(-frac{z_i e psi}{k_B T}right)]where (psi) is the electrostatic potential, (rho) is the charge density, (epsilon) is the permittivity of the medium, (z_i) is the valence of ion (i), (e) is the elementary charge, (c_i) is the concentration of ion (i), (k_B) is the Boltzmann constant, and (T) is the temperature.## Step 4: Debye-Huckel ApproximationFor low potentials or high temperatures, the exponential term in the Poisson-Boltzmann equation can be approximated using the Debye-Huckel approximation, which linearizes the equation:[expleft(-frac{z_i e psi}{k_B T}right) approx 1 - frac{z_i e psi}{k_B T}]This simplification leads to a linearized Poisson-Boltzmann equation, which can be solved to obtain the electrostatic potential distribution around the charged body.## Step 5: Behavior of Sodium IonsUsing the Debye-Huckel approximation, the electrostatic potential around a negatively charged body in a NaCl solution can be described by the equation:[nabla^2 psi = kappa^2 psi]where (kappa) is the Debye-Huckel parameter, which depends on the ionic strength of the solution and the temperature. The solution to this equation gives the potential distribution as a function of distance from the charged body. The positive sodium ions are attracted to the negatively charged body and accumulate near its surface, creating an electric double layer.## Step 6: ConclusionIn conclusion, the behavior of positive sodium ions in relation to a negatively charged body in a NaCl solution can be described by considering the electrostatic potential and the role of mobile ions. The Poisson-Boltzmann equation and the Debye-Huckel approximation provide a theoretical framework for understanding this behavior, showing how the sodium ions are attracted to the negatively charged body and form an electric double layer.The final answer is: boxed{The sodium ions accumulate near the surface of the negatively charged body.}

🔑:Beats are audible at lower frequencies when listening through headphones due to the physiological mechanisms involved in sound localization and the limitations imposed by the frequency of sound on phase detection. To understand this phenomenon, let's dive into the details of sound localization and the role of the human auditory system.Sound LocalizationSound localization is the ability to determine the source of a sound in space. This is achieved through the detection of interaural differences in time and intensity between the sounds arriving at each ear. The human auditory system uses these differences to calculate the location of the sound source. There are two primary cues for sound localization:1. Interaural Time Difference (ITD): The difference in time between the sound arriving at each ear. This cue is more effective for low-frequency sounds (< 1.5 kHz).2. Interaural Level Difference (ILD): The difference in intensity between the sound arriving at each ear. This cue is more effective for high-frequency sounds (> 1.5 kHz).Physiological MechanismsThe human auditory system detects and processes sound through a complex series of mechanisms involving the ear, auditory nerve, and brain. The key components involved in sound localization are:1. Basilar Membrane: A flexible, spiral-shaped structure in the cochlea that vibrates in response to sound waves. The basilar membrane is tonotopically organized, meaning that different regions respond to different frequencies.2. Hair Cells: Specialized sensory cells on the basilar membrane that convert mechanical vibrations into electrical signals.3. Auditory Nerve: The nerve that transmits these electrical signals to the brain.Phase DetectionWhen two sounds with slightly different frequencies are presented to the listener, they produce a beating effect, which is the periodic variation in amplitude. The human auditory system can detect these beats through the phase differences between the two sounds. The phase information is detected by the hair cells on the basilar membrane, which respond to the timing differences between the sound waves.At lower frequencies, the wavelength of the sound is longer, and the phase differences between the two sounds are more pronounced. This makes it easier for the auditory system to detect the beats. In contrast, at higher frequencies, the wavelength is shorter, and the phase differences are less pronounced, making it more difficult to detect the beats.Limitations Imposed by FrequencyThe frequency of sound imposes limitations on phase detection. At high frequencies, the auditory system has difficulty detecting phase differences due to the shorter wavelength and the limited resolution of the basilar membrane. This is known as the "ambiguity problem." As a result, the auditory system relies more on intensity differences (ILD) for sound localization at high frequencies.In contrast, at low frequencies, the auditory system can detect phase differences more easily, making it possible to localize sounds based on ITD. This is why beats are more audible at lower frequencies when listening through headphones.Headphone ListeningWhen listening to sound through headphones, the interaural differences in time and intensity are preserved, allowing the auditory system to localize the sound source. However, the headphones can also introduce artifacts, such as interaural crosstalk, which can affect sound localization. At lower frequencies, the beating effect is more pronounced, making it easier to detect the phase differences and localize the sound source.In summary, the human auditory system detects and processes sound localization through a complex series of mechanisms involving the basilar membrane, hair cells, and auditory nerve. The phase information is detected by the hair cells, and the limitations imposed by the frequency of sound affect the ability to detect phase differences. At lower frequencies, the auditory system can detect phase differences more easily, making beats more audible when listening through headphones.

🔑:## Step 1: Understand the Kerr-Newman black hole parametersThe Kerr-Newman black hole is characterized by three parameters: mass (M), charge (Q), and angular momentum (J). The extremal condition for a Kerr-Newman black hole is given by (M^2 = Q^2 + J^2/M^2), which simplifies to (M^2 = Q^2 + (J/M)^2) for an extremal black hole, implying a balance between the mass, charge, and angular momentum.## Step 2: Consider the process of adding charge to achieve an extremal solutionTo achieve an extremal black hole solution where (M = Q), one would need to add charge to the black hole. The addition of charge (dQ) from infinity to the horizon of the black hole would increase its total charge. However, this process must be considered in the context of the black hole's energy and the constraints imposed by the extremal condition.## Step 3: Implications on angular momentum and angular velocityAs charge is added to the black hole, its angular momentum (J) and angular velocity (Omega) could be affected. The relationship between (J), (M), and (Q) in an extremal black hole suggests that if (M = Q), then (J) must be zero to satisfy the extremal condition (M^2 = Q^2 + (J/M)^2), since (M^2 = Q^2) leaves no room for a non-zero (J^2/M^2) term without violating the extremal condition.## Step 4: Analyze the effect on the Kerr-Newman solutionThe Kerr-Newman solution describes a rotating, charged black hole. Achieving an extremal condition where (M = Q) would imply a non-rotating ((J = 0)) black hole, as any rotation (non-zero (J)) would prevent the black hole from being extremal if (M = Q). This is because the extremal condition with (M = Q) does not allow for any additional contribution from angular momentum without exceeding the extremal limit.## Step 5: Conclusion on achieving an extremal black holeTo achieve an extremal black hole solution where (M = Q), the black hole must not rotate ((J = 0)), implying (Omega = 0) since the angular velocity (Omega) is directly related to (J) and (M). This condition simplifies the Kerr-Newman solution to the Reissner-Nordström solution, which describes a non-rotating, charged black hole.The final answer is: boxed{0}

🔑:## Step 1: Define the problem in terms of relativistic kinematicsWe start with a proton (p) colliding with a neutron (n) at rest, resulting in the formation of a deuterium nucleus (d) and the emission of a photon (γ). The reaction can be written as p + n → d + γ. We need to find the wavelength of the photon (λ) as a function of the proton's initial momentum (p) and the angle (θ) between the photon's direction and the proton's initial direction.## Step 2: Apply conservation of energy and momentumConservation of energy and momentum in relativistic kinematics involves considering the four-momentum of each particle. The four-momentum of a particle is given by (E, p), where E is the total energy of the particle and p is its momentum. For the proton, the initial four-momentum is (E_p, p_p), for the neutron at rest it's (m_n, 0), for the deuterium nucleus it's (E_d, p_d), and for the photon, it's (E_γ, p_γ).## Step 3: Calculate the energy of the protonThe energy of the proton (E_p) can be calculated using the relativistic energy-momentum equation E^2 = (pc)^2 + (mc^2)^2, where m is the rest mass of the proton, p is its momentum, and c is the speed of light. For the proton, E_p = sqrt((p_p*c)^2 + (m_p*c^2)^2).## Step 4: Consider the neutron's energyThe neutron's energy (E_n) is simply its rest mass energy since it's at rest, so E_n = m_n*c^2.## Step 5: Calculate the energy of the deuterium nucleusAfter the collision, the deuterium nucleus will have a certain energy (E_d) and momentum (p_d). The energy of the deuterium can be found from the conservation of energy: E_p + E_n = E_d + E_γ.## Step 6: Calculate the photon's energy and momentumThe photon's energy (E_γ) is related to its momentum (p_γ) by E_γ = p_γ*c. The energy of the photon can also be found from the conservation of energy equation: E_γ = E_p + E_n - E_d.## Step 7: Apply conservation of momentumConservation of momentum gives us p_p = p_d + p_γ. Since the photon's momentum is related to its energy by p_γ = E_γ / c, we can express the conservation of momentum in terms of energies and the angle θ.## Step 8: Express the deuterium nucleus's energy in terms of its momentumThe deuterium nucleus's energy can be expressed as E_d = sqrt((p_d*c)^2 + (m_d*c^2)^2), where m_d is the rest mass of the deuterium nucleus.## Step 9: Use the conservation laws to find the photon's energyBy combining the conservation of energy and momentum, and considering the relativistic energies, we can solve for the photon's energy (E_γ) in terms of the proton's initial momentum (p_p) and the angle (θ).## Step 10: Find the wavelength of the photonThe wavelength (λ) of the photon is related to its energy (E_γ) by λ = hc / E_γ, where h is Planck's constant.The final answer is: boxed{lambda = frac{hc}{E_p + m_n c^2 - sqrt{(E_p + m_n c^2)^2 - (m_d c^2)^2 + (p_p c)^2 - 2 p_p c sqrt{(E_p + m_n c^2)^2 - (m_d c^2)^2} cos(theta)}}}