Electrostatic energy of an infinite range of charges. A) Two spherical ions with charges Q1 and Q2 and radii R1 and R2 are inside an aqueous solution. Write an expression for the ratio of the probabilities that the two ions are at infinite distance from each other (r = + \infty) or in contact (r = rmin R1 + R2).
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Exercise 1 : Electrostatic energy of an infinite range of charges. A) Two spherical ions with charges Q1 and Q2 and radii R1 and R2 are inside an aqueous solution. Write an expression for the ratio of the probabilities that the two ions are at infinite distance from each other (r = + \infty) or in contact (r = rmin R1 + R2).
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- help me with part e please9 V B. The circuit at right is made from an ideal battery connected to 4 ideal resistors as shown. i. What is the total resistance of this circuit? Show your work. R4= 352 R1 = 152 R3 = 452 ii. Find the current through each of the 4 resistors. Show your work. R2= 752 iii. Rank the potential difference across the 4 resistors from greatest to least. Explain your reasoning. iv. If resistor R3 is cut out of the circuit, without replacing it with anything, will the current through resistor R2 increase, decrease, or stay the same? Explain your answer or show your work.Part B: Error propagation Consider that you measured an angle 0 to be 40 degrees. The uncertainty of your measurement is + 1 degree. Using the rules for error propagations, determine the uncertainty of sin(0). Clearly show your work. Hint: The uncertainty of any function of a single variable, z = f(x), is the derivative of the function calculated at that point multiplied by the uncertainty of the variable. A(z) df Ax dx
- Self-test 1B.1 Derive the expression for (v²) in eqn 1B.7 by evaluating the integral in eqn 1B.6 with n = 2.Solve part- a& bProblem B.2 Each of the ODES shown below is second order in y, with y1 as a solution. Reduce the ODE from being second order in y to being first order in w, with w being the only response variable appearing in the ODE. Combine like terms. Show your work. B.2.a. xy" –y' +y = 0 Y1 = x* В.2.b. y" + 9у %3D 0 sin(3t) B.2.c. x²y" +y = 0 Y1 = x²/3
- A thin spherical shell with radius R₁ = 4.00 cm is concentric with a larger thin spherical shell with radius R2 = 6.00 cm. Both shells are made of insulating material. The smaller shell has charge q₁ = +6.00 nC distributed uniformly over its surface, and the larger shell has charge q2 = - 9.00 nC distributed uniformly over its surface. Take the electric potential to be zero at an infinite distance from both shells. - For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Potential vsurface and field magnitude esurface at the surface. ▶ Part A Part BExercise b. Hard spin glasses.Physicists use a model called an Ising spin glass to study magnetic materials. We havea graph G = (V, E) and a function J : E → Z, where J(u, v) is the interaction strengthof edge {u, v} ∈ E. A state is a function s : V → {−1, +1}.1 The energy of a states is H(s) = −P{u,v}∈EJ(u, v) · s(u) · s(v).2 An edge {u, v} is called ferromagnetic ifJ(u, v) > 0, and antiferromagnetic if J(u, v) < 0. Ferromagnetic edges “pressure” uand v to be the same spin, and antiferromagnetic edges “pressure” them to be different.We wish to find the ground state, i.e., values of s(u) for each u ∈ V that minimize theenergy. Phrased as a decision problem, we ask whether a state exists below a certainenergy threshold:Spin-GlassInput: a graph G = (V, E), interaction strengths J : E → Z, and a threshold hˆ ∈ ZQuestion: is there a state s : V → {−1, +1} with H(s) ≤ hˆ?Show that Spin-Glass is NP-complete.1Each s(u) is called the spin of u; if u is a magnetic domain, think of s(u) as…