🔑:To find the torque acting on the coil about the hinge line, we first need to determine the magnetic moment of the coil and then calculate the torque using the formula for torque due to a magnetic field.## Step 1: Determine the magnetic moment of the coil.The magnetic moment (μ) of a coil can be calculated using the formula μ = NIA, where N is the number of turns, I is the current, and A is the area of the coil. The area of the coil is given by length times width, so A = 10 cm * 5.0 cm = 0.10 m * 0.050 m = 0.0050 m^2. Therefore, μ = 20 * 0.10 A * 0.0050 m^2 = 0.010 m^2*A.## Step 2: Express the magnetic moment in unit-vector notation.Since the coil is mounted in the x-y plane at an angle θ = 30° to the direction of the magnetic field, we need to express the magnetic moment in terms of its components. The magnetic moment vector μ can be written as μ = μx*i + μy*j, where i and j are unit vectors in the x and y directions, respectively. Given that the magnetic field is directed along the x-axis and the coil is at 30° to this field, the magnetic moment components can be found using μx = μ*cos(30°) and μy = μ*sin(30°).## Step 3: Calculate the components of the magnetic moment.Using the value of μ from Step 1, we calculate μx and μy. μx = 0.010 m^2*A * cos(30°) and μy = 0.010 m^2*A * sin(30°). Since cos(30°) = √3/2 and sin(30°) = 1/2, μx = 0.010 * √3/2 = 0.00866 m^2*A and μy = 0.010 * 1/2 = 0.0050 m^2*A.## Step 4: Calculate the torque acting on the coil.The torque (τ) due to a magnetic field (B) is given by τ = μ x B, where "x" denotes the cross product. Since the magnetic field is directed along the x-axis, B = Bx*i. The cross product μ x B will only have a component in the z-direction because the coil is in the x-y plane. The magnitude of the torque is τ = μy*Bx, since the cross product of μ and B in this case simplifies to the product of the y-component of μ and the x-component of B.## Step 5: Calculate the numerical value of the torque.Given B = 0.50 T and using μy from Step 3, τ = 0.0050 m^2*A * 0.50 T = 0.0025 N*m.## Step 6: Express the torque in unit-vector notation.The torque vector τ is directed along the z-axis (since it's the result of a cross product involving vectors in the x-y plane), so τ = τz*k, where k is the unit vector in the z-direction. Since we found τ = 0.0025 N*m, τ = 0.0025 N*m * k.The final answer is: boxed{0.0025k}

🔑:A classic problem in electrical engineering!Let's break it down step by step.Given:* A simple LC circuit with a capacitor (C) and an inductor (L) connected in series.* The capacitor is initially charged to a maximum charge Qmax.* The circuit is closed, and we want to find the charge on the capacitor as a function of time, Q(t).Kirchhoff's Laws:We'll use Kirchhoff's voltage law (KVL) to derive the relationship for Q(t). KVL states that the sum of voltage changes around a closed loop is zero.Electric Field Concept:The electric field (E) across the capacitor is related to the charge (Q) and capacitance (C) by:E = Q/CThe voltage (V) across the capacitor is the line integral of the electric field:V = ∫E dx = Q/CSimilarly, the voltage across the inductor (L) is related to the current (I) and inductance (L) by:V = L(dI/dt)Derivation:Applying KVL to the LC circuit, we get:V_C + V_L = 0Substituting the expressions for V_C and V_L, we get:Q/C + L(dI/dt) = 0Now, we need to express the current (I) in terms of the charge (Q). Since the current is the rate of change of charge, we can write:I = dQ/dtSubstituting this into the previous equation, we get:Q/C + L(d^2Q/dt^2) = 0Rearranging the terms, we get:L(d^2Q/dt^2) + Q/C = 0This is a second-order linear homogeneous differential equation with constant coefficients.Solution:The solution to this differential equation is a sinusoidal function:Q(t) = Qmax cos(ωt + φ)where ω is the angular frequency, φ is the phase angle, and Qmax is the maximum charge.To find ω, we can use the fact that the differential equation has a sinusoidal solution. Substituting Q(t) = Qmax cos(ωt + φ) into the differential equation, we get:-Lω^2 Qmax cos(ωt + φ) + Qmax cos(ωt + φ)/C = 0Simplifying and rearranging, we get:ω^2 = 1/LCω = 1/√(LC)This is the angular frequency of the LC circuit.Final Answer:The charge on the capacitor as a function of time is:Q(t) = Qmax cos(ωt + φ)where ω = 1/√(LC) is the angular frequency, and φ is the phase angle.Note that the phase angle φ depends on the initial conditions of the circuit. If the capacitor is initially charged to Qmax, then φ = 0, and the solution simplifies to:Q(t) = Qmax cos(ωt)This is the classic sinusoidal solution for a simple LC circuit.

🔑:## Step 1: Understanding the Plasma StateA plasma is a high-energy state of matter where atoms or molecules are ionized, meaning they have lost or gained electrons to form ions. In the case of ionized hydrogen gas, we have protons (positively charged) and electrons (negatively charged). The temperature of 1 keV (kilo-electronvolt) is extremely high, equivalent to about 11.6 million degrees Celsius. At such temperatures, the kinetic energy of the particles is very high, and the plasma behaves as a collection of charged particles interacting through electromagnetic forces.## Step 2: Neutrality of the PlasmaDespite the high energy and the presence of charged particles, plasmas tend to remain neutral overall. This neutrality is maintained because the number of positive charges (protons) equals the number of negative charges (electrons) over a sufficiently large volume, known as the Debye sphere. The Debye length is a measure of the distance over which electric fields are shielded by the plasma, and within this distance, the plasma can be considered neutral.## Step 3: Role of Coulomb ForcesCoulomb forces between protons and electrons play a crucial role in maintaining plasma neutrality. The Coulomb force is an electrostatic force that acts between charged particles. In a plasma, the attractive force between protons and electrons helps to maintain the neutrality by preventing the separation of charges over large distances. However, at the high temperatures of a plasma, the kinetic energy of the particles often exceeds the potential energy of the Coulomb interaction, allowing the plasma to behave more like a gas of charged particles than a collection of atoms or molecules.## Step 4: Electron MobilityElectron mobility is the ability of electrons to move freely within the plasma. At high temperatures, electrons have high velocities and can move rapidly in response to electric fields. This mobility is crucial for maintaining plasma neutrality because it allows electrons to quickly move to areas where there is an excess of positive charge, thus neutralizing the plasma. The high mobility of electrons in a plasma is one reason why plasmas can maintain their neutrality despite the intense kinetic energy of the particles.## Step 5: Effects of Magnetic FieldsMagnetic fields can significantly affect plasma confinement and behavior. In the presence of a magnetic field, charged particles (both protons and electrons) follow helical paths around the magnetic field lines. This can lead to the confinement of the plasma, as the particles are prevented from moving freely in all directions. Magnetic confinement is a key principle behind many plasma devices, including tokamaks, which are designed to achieve controlled nuclear fusion. The magnetic field does not directly affect the neutrality of the plasma but can influence the distribution of charges and the overall stability of the plasma.## Step 6: Conclusion on Neutrality and Coulomb ForcesIn conclusion, the likelihood of the plasma remaining neutral at a temperature of 1 keV is high due to the balance between the number of positive and negative charges over a large volume. The Coulomb forces between protons and electrons contribute to this neutrality by attracting opposite charges and preventing large-scale charge separation. Electron mobility and the effects of magnetic fields on plasma confinement also play critical roles in the behavior and stability of the plasma.The final answer is: boxed{1}

🔑:## Step 1: Understand the dimensionality of Newton's constant G in a 4D theoryIn a 4D theory, Newton's constant G has dimensions of [L^3 M^{-1} T^{-2}] or [L^2][M^{-1}] when considering the gravitational force between two masses, where L is length, M is mass, and T is time. This is derived from the gravitational force equation F = G frac{m_1 m_2}{r^2}.## Step 2: Determine the relationship between the gravitational coupling constant and Newton's constant GThe gravitational coupling constant, often represented in the context of quantum field theory or particle physics as kappa or in relation to the Planck mass M_P, is related to Newton's constant G. Specifically, in the context of General Relativity and its quantization, the gravitational coupling constant can be considered as kappa = sqrt{8pi G}, where G is Newton's constant. This relationship arises from the Einstein field equations when considering the coupling of gravity to matter fields.## Step 3: Discuss the dimensionality of the gravitational coupling constantGiven that kappa = sqrt{8pi G}, and knowing the dimensions of G are [L^3 M^{-1} T^{-2}], we can infer the dimensions of kappa. Since kappa is a square root of G, its dimensions would be sqrt{[L^3 M^{-1} T^{-2}]} = [L^{3/2} M^{-1/2} T^{-1}]. However, in the context of quantum field theory, coupling constants are often discussed in terms of their dimensionality in relation to the energy or mass scales. The dimensionality of kappa in terms of mass or energy is what's crucial for discussing renormalizability.## Step 4: Analyze the effect of dimensionality on the renormalizability of General RelativityIn quantum field theory, the renormalizability of a theory is closely related to the dimensionality of its coupling constants. A theory is considered renormalizable if its coupling constants are dimensionless or have positive mass dimension. The gravitational coupling constant kappa has a negative mass dimension (-1 in 4D when considering it in terms of energy or mass units, since G has dimensions of [L^2] or [M^{-2}] when considering c = 1 and hbar = 1). This negative dimensionality indicates that General Relativity, as a quantum field theory, is non-renormalizable because the coupling constant's dimension leads to infinities that cannot be absorbed by counterterms in a straightforward manner, unlike in renormalizable theories.The final answer is: boxed{kappa = sqrt{8pi G}}