4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant,...
Q3. [22 marks] The Dirichlet's problem for a disc of radius a is stated as follows: r(a, θ)-/(0) for osas2m, where the function f (0) is integrable [10 marks] Find the general solution of u(r, θ) (i) (7 marks] if f (θ)-sin|-θ | , find the specific solution u (r,0) (ii) [ (ii) [5 marks] Use the solution in (ii) to deduce that 4n1-9) 18 Q4. [24marks] Consider the second order linear partial differential equation Q3. [22 marks] The Dirichlet's...
The potential energy of two atoms in a diatomic molecule is approximated by U(r)=d/r12-b/r6, where r is the spacing between atoms and a and b are positive constants. Find the force F(r) on one atom as a function of r. Find the equilibrium distance between the two atoms. Express your answer in terms of the variables a and b. Is this equilibrium stable? Suppose the distance between the two atoms is equal to the equilibrium distance found in part (b)....
6. (a) Find the directional derivative of f(, y,z)at Po(1,10 in the direction of u-i-2j+k. 4 marks] (b) Letz201), where f is a differentiable function, show that 08z 4 marks] (c) Use Lagrange Multiplier to find the absolute maximum and minimum of 10 (x,y)-x +y subject to 2 12 marks 6. (a) Find the directional derivative of f(, y,z)at Po(1,10 in the direction of u-i-2j+k. 4 marks] (b) Letz201), where f is a differentiable function, show that 08z 4 marks]...
4. Rank-2 tensors A charge q moves with a constant velocity tensor rby rHv 2 u. Define the antisymmetric ) where r" is the four velocity of the charge. Further define u (n" =-rH"ru/2. (x-ut)2 1-u2/c2 + y2 + z2 |(a) Show that r2 (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the x,y or z component of E 4TC (r2)3/2 Where the 4. Rank-2 tensors A charge q moves...
4. Rank-2 tensors A charge q moves with a constant velocity tensor rby rHv 2 u. Define the antisymmetric ) where r" is the four velocity of the charge. Further define u (n" =-rH"ru/2. (x-ut)2 1-u2/c2 + y2 + z2 |(a) Show that r2 (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the x,y or z component of E 4TC (r2)3/2 Where the 4. Rank-2 tensors A charge q moves...
4. Rank-2 tensors A charge q moves with a constant velocity tensor rby rHv 2 u. Define the antisymmetric ) where r" is the four velocity of the charge. Further define u (n" =-rH"ru/2. (x-ut)2 1-u2/c2 + y2 + z2 |(a) Show that r2 (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the x,y or z component of E 4TC (r2)3/2 Where the 4. Rank-2 tensors A charge q moves...
4. Rank-2 tensors A charge q moves with a constant velocity tensor rby rHv 2 u. Define the antisymmetric ) where r" is the four velocity of the charge. Further define u (n" =-rH"ru/2. (x-ut)2 1-u2/c2 + y2 + z2 |(a) Show that r2 (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the x,y or z component of E 4TC (r2)3/2 Where the 4. Rank-2 tensors A charge q moves...
4. Rank-2 tensors A charge q moves with a constant velocity tensor rby rHv 2 u. Define the antisymmetric ) where r" is the four velocity of the charge. Further define u (n" =-rH"ru/2. (x-ut)2 1-u2/c2 + y2 + z2 |(a) Show that r2 (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the x,y or z component of E 4TC (r2)3/2 Where the 4. Rank-2 tensors A charge q moves...
4. Rank-2 tensors A charge q moves with a constant velocity u ux. Define the antisymmetric tensor r" by r"-1 (ημ2-η"z"), where ημ įs the four velocity of the charge. Further define (a) Show that r-lauP + y2 + z2. -tu (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the s,y or z component of E where the Imc (r2) TC 4. Rank-2 tensors A charge q moves with a...
9. (8 marks) Let F be an inverse force field given by k F(r,y,z) r13 r, where k is a constant. Find the work done by F on an object as its point of application moves along the -axis from A(1,0, 0) to B(2,0,0) 9. (8 marks) Let F be an inverse force field given by k F(r,y,z) r13 r, where k is a constant. Find the work done by F on an object as its point of application moves...