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11. One-dimensional polymer, consider a polymer formed by connecting N disc-shaped molecules into a one-dimensional chain....
1) A simple picture of a polymer chain in two dimensions is designed as a single...
1) A simple picture of a polymer chain in two dimensions is designed as a single long chain of linking N monomers of the length of a. Monomers are linked to together with chemical reaction. The monomers can be oriented in 4 directions given by +ai, +aſ,+aj and – aj A constant force is applied to expand the polymer along +x direction only. The energy of each monomer is given by x = -å. F where F = Fi is...
5. Consider a polymer consisting of N links (like links of a chain) of length L....
5. Consider a polymer consisting of N links (like links of a chain) of length L. The links are massless. One end of the polymer is fixed at height z = 0. The other end is attached to a mass m. Each link can pivot about its attachment with the previous link or the fixed end so that a link may carry the polymer "up" or "down" a distance L. For example, the figure shows a polymer containing four links,...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and...
2. Microcanonical ensemble: One-dimensional chain. (24 pts.) Consider a one-dimensional chain consisting of N segments as illus- trated in Figure 1. Let the length of each segment be a when the...
2. Microcanonical ensemble: One-dimensional chain. (24 pts.) Consider a one-dimensional chain consisting of N segments as illus- trated in Figure 1. Let the length of each segment be a when the long dimension of the segment is parallel to the chain and 0 when the long dimension is normal to the chain direction. Each segment has just two non-degenerate states: long dimension parallel to the chain or perpen- dicular to the chain. Now consider a macrostate of the chain in...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic moment μ The spins are kept in a magnetic field with magnetic induction B along the z direction. The spin can point either up, t, or down, , relative to the z axis. The energy of particle with spin down is e B and that of particle with spin up is ε--B. We assume that the system is isolated from. its environment so...
QUESTION 5 A angular coil of wire H 24 c n by L-34 c and car yng a cu ent of 1.6 A İs one ted with the plane of its loop pe pendicular to a uniform 1.6 T magnetic field as shown in the figure. T...
QUESTION 5 A angular coil of wire H 24 c n by L-34 c and car yng a cu ent of 1.6 A İs one ted with the plane of its loop pe pendicular to a uniform 1.6 T magnetic field as shown in the figure. The coil is then rotated through a 28° angle into the plane. Calculate the torque that the magnetic field exerts on the coil. Hint: In order to help visualize this three-dimensional problem, make a...
1) A simple picture of a polymer chain in two dimensions is designed as a single long chain of linking N monomers of the length of a. Monomers are linked to together with chemical reaction. The monomers can be oriented in 4 directions given by +ai, +aſ,+aj and – aj A constant force is applied to expand the polymer along +x direction only. The energy of each monomer is given by x = -å. F where F = Fi is...
5. Consider a polymer consisting of N links (like links of a chain) of length L. The links are massless. One end of the polymer is fixed at height z = 0. The other end is attached to a mass m. Each link can pivot about its attachment with the previous link or the fixed end so that a link may carry the polymer "up" or "down" a distance L. For example, the figure shows a polymer containing four links,...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and...
2. Microcanonical ensemble: One-dimensional chain. (24 pts.) Consider a one-dimensional chain consisting of N segments as illus- trated in Figure 1. Let the length of each segment be a when the long dimension of the segment is parallel to the chain and 0 when the long dimension is normal to the chain direction. Each segment has just two non-degenerate states: long dimension parallel to the chain or perpen- dicular to the chain. Now consider a macrostate of the chain in...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic moment μ The spins are kept in a magnetic field with magnetic induction B along the z direction. The spin can point either up, t, or down, , relative to the z axis. The energy of particle with spin down is e B and that of particle with spin up is ε--B. We assume that the system is isolated from. its environment so...
QUESTION 5 A angular coil of wire H 24 c n by L-34 c and car yng a cu ent of 1.6 A İs one ted with the plane of its loop pe pendicular to a uniform 1.6 T magnetic field as shown in the figure. The coil is then rotated through a 28° angle into the plane. Calculate the torque that the magnetic field exerts on the coil. Hint: In order to help visualize this three-dimensional problem, make a...
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