🔑:Given,Height of inclined plane, h = 3 mAngular velocity, ω = 2√2 rad/sAcceleration due to gravity, g = 10 m/s2As we know that,ω2 = (2gh) / r2⇒ r2 = (2gh) / ω2⇒ r2 = (2×10×3) / (2√2)2⇒ r2 = (60) / (2×2)⇒ r2 = 15⇒ r = √15⇒ r = 3.87 mHence, the radius of the cylinder is 3.87 m.

🔑:The AdS/CFT correspondence, also known as the Maldacena duality, is a theoretical framework in physics that relates a conformal field theory (CFT) on the boundary of a spacetime to a gravitational theory in the bulk. One of the fascinating aspects of this correspondence is the concept of non-locality, which arises from the interplay between the locality of the CFT on the boundary and the non-locality observed in the bulk.Locality in the CFTIn the CFT, locality refers to the fact that the theory is defined in terms of local operators, which are functions of the spacetime coordinates on the boundary. These operators satisfy the usual rules of quantum field theory, including the commutator algebra and the notion of causality. The CFT is a local theory, meaning that the correlation functions between operators at different points on the boundary decay rapidly as the distance between the points increases.Non-locality in the bulkIn contrast, the gravitational theory in the bulk, such as string theory or supergravity, exhibits non-locality. This non-locality arises from the fact that the bulk theory is a theory of gravity, which is inherently non-local due to the presence of gravitational waves and the curvature of spacetime. The non-locality in the bulk is reflected in the fact that the correlation functions between operators at different points in the bulk do not decay rapidly with distance, but instead exhibit a power-law behavior.Relating locality and non-localityThe AdS/CFT correspondence provides a way to relate the locality of the CFT on the boundary to the non-locality observed in the bulk. The key idea is that the CFT on the boundary is a holographic representation of the bulk theory, meaning that the information contained in the CFT is encoded in a way that is not local in the bulk. Specifically, the CFT operators on the boundary are related to the bulk operators through a non-local transformation, known as the bulk-to-boundary propagator.This non-local transformation maps the local operators on the boundary to non-local operators in the bulk, which are sensitive to the global structure of the spacetime. The non-locality of the bulk theory is thus encoded in the CFT through the non-local transformation, which allows us to reconstruct the bulk theory from the CFT.Implications for quantum gravity and the holographic principleThe AdS/CFT correspondence and the concept of non-locality have far-reaching implications for our understanding of quantum gravity and the holographic principle. Some of the key implications include:1. Holographic principle: The AdS/CFT correspondence provides a concrete realization of the holographic principle, which states that the information contained in a region of spacetime can be encoded on the surface of that region. The non-locality of the bulk theory is a key aspect of this principle, as it allows us to reconstruct the bulk theory from the boundary theory.2. Quantum gravity: The AdS/CFT correspondence provides a new perspective on quantum gravity, which is a theory that seeks to merge quantum mechanics and general relativity. The non-locality of the bulk theory is a key feature of quantum gravity, and the AdS/CFT correspondence provides a way to understand how this non-locality arises from the locality of the CFT on the boundary.3. Black hole physics: The AdS/CFT correspondence has been used to study black hole physics, including the behavior of black holes in the presence of a heat bath. The non-locality of the bulk theory plays a key role in understanding the thermalization of black holes and the behavior of black hole entropy.4. Cosmology: The AdS/CFT correspondence has also been used to study cosmology, including the evolution of the universe and the formation of structure. The non-locality of the bulk theory is important for understanding the large-scale structure of the universe and the distribution of matter and energy.In conclusion, the concept of non-locality in the context of the AdS/CFT correspondence is a rich and fascinating area of research that has far-reaching implications for our understanding of quantum gravity, the holographic principle, and the behavior of black holes and the universe as a whole. The interplay between the locality of the CFT on the boundary and the non-locality observed in the bulk provides a new perspective on the nature of spacetime and the behavior of matter and energy at the most fundamental level.

🔑:## Step 1: Understanding the energy-momentum tensorThe energy-momentum tensor is a fundamental concept in field theory, describing the distribution of energy and momentum in a system. It is derived from the Lagrangian of the system, which encodes the dynamics of the fields involved.## Step 2: Symmetry requirement for the energy-momentum tensorFor the energy-momentum tensor to be physically meaningful, it must be symmetric. This symmetry is crucial because it ensures that the tensor can be properly conserved, reflecting the conservation of energy and momentum in the system.## Step 3: Deriving the energy-momentum tensor from a LagrangianThe energy-momentum tensor (T_{munu}) can be derived from a Lagrangian (mathcal{L}) using the formula (T_{munu} = frac{partial mathcal{L}}{partial (partial^mu phi)} partial_nu phi - eta_{munu} mathcal{L}) for a scalar field (phi), where (eta_{munu}) is the Minkowski metric. For more complex Lagrangians, such as those involving vector fields (like the Maxwell Lagrangian), the derivation must be adjusted accordingly.## Step 4: Ensuring symmetry in the energy-momentum tensorTo ensure the energy-momentum tensor is symmetric, one must consider the Lagrangian's dependence on the metric. If the Lagrangian does not explicitly contain the metric, it may be necessary to rewrite it to include the correct metric dependence. This is often achieved by considering how the fields transform under coordinate transformations and ensuring that the action (the integral of the Lagrangian over spacetime) is invariant.## Step 5: Rewriting the Lagrangian for metric dependenceFor a Free boson Lagrangian (mathcal{L} = frac{1}{2} partial_mu phi partial^mu phi - frac{1}{2} m^2 phi^2), the metric dependence is implicit in the contraction of indices. To make this dependence explicit and ensure the energy-momentum tensor is symmetric, one considers the Lagrangian in a general form that includes the metric, (mathcal{L} = frac{1}{2} g^{munu} partial_mu phi partial_nu phi - frac{1}{2} m^2 phi^2), where (g^{munu}) is the metric tensor.## Step 6: Physical significance of a symmetric energy-momentum tensorA symmetric energy-momentum tensor is crucial for the conservation of angular momentum and for ensuring that the theory respects the principles of general relativity, where the energy-momentum tensor acts as the source of the gravitational field. In the context of field theories like the Free boson Lagrangian or the Maxwell Lagrangian, a symmetric energy-momentum tensor ensures that energy, momentum, and angular momentum are properly conserved, which is fundamental for the predictive power and consistency of these theories.The final answer is: boxed{0}

🔑:## Step 1: Understanding Intrinsic SemiconductorsIntrinsic semiconductors are materials that have electrical conductivity between that of a conductor and an insulator. This conductivity is due to the thermal excitation of electrons from the valence band to the conduction band, creating pairs of electrons and holes. Electrons are negatively charged, while holes (the absence of an electron in a position where one could exist) are positively charged.## Step 2: Charge Carrier RecombinationWhen an electron in the conduction band recombines with a hole in the valence band, the electron fills the "hole" or the empty state in the valence band. This process is known as recombination. During recombination, the energy released can be in the form of heat (phonons) or light (photons), depending on the material and the conditions.## Step 3: Effect on Total ChargeSince electrons and holes have opposite charges, when they recombine, the net effect is the annihilation of both the positive and negative charge carriers. This means that the total charge of the material remains unchanged because the number of positive charges (holes) and negative charges (electrons) decreases equally.## Step 4: Scenario with Injected Electrons and HolesIf 50 electrons and 50 holes are injected into the material, these charge carriers will move freely within the semiconductor. Due to thermal energy, electrons and holes will randomly collide and recombine. Since the number of injected electrons equals the number of injected holes, every electron can potentially recombine with a hole.## Step 5: Final State of the MaterialAfter all the injected electrons and holes have recombined, the material returns to its original state before the injection. The recombination process does not alter the material's electrical neutrality because each electron that recombines with a hole effectively cancels out both the positive and negative charge, leaving no net charge imbalance.## Step 6: Electrical NeutralityThe electrical neutrality of the material is maintained throughout the process. Initially, the material is electrically neutral. The injection of equal numbers of electrons and holes introduces equal amounts of negative and positive charge, but after recombination, the material returns to its neutral state, as all the injected charge carriers have been eliminated.The final answer is: boxed{0}