Some force fields in physics have the form..., where "k" is a constant , "r" is the position vector. These fields come from the gradient of ´ a scalar field p(x). Determine this scalar field.
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We know from electrostatics that if we have a scalar electrostatic potential V, then there exists an electric field that satisfies: Of course, not all vector fields can be written as the gradient of a scalar function. (a) Show that the electric field given below is not the result of an electrostatic potential. E(x, y, z) = ( 3.0m,2 ) ( yi-TJ (b) Just because this electric field can't come from an electrostatic potential, it doesn't mean it can't exist...
4. We know from electrostatics that if we have a scalar electrostatic potential V, then there exists an electric field that satisfies: Of course, not all vector fields can be written as the gradient of a scalar function. (a) Show that the electric field given below is not the result of an electrostatic potential (b) Just because this electric field can't come from an electrostatic potential, it doesn't mean it can't exist - it just can't be created by static...
On the figures a-d, three different vector fields are drawn ラ a) Write expressions for the fields u(x)- (u(x, y), v(x, y)) in a fitting cartesian coordinatsystem b) Calculate the divergense (V-u) and rotation (Vx u) in the four instances. On the figures below we have two different scalar fields for example representing pressure field. 1000 005 c) Find an expression for the field p(x, y) d) Calculate and draw the gradient field in the two instances On the figures...
Learning Goal: To gain insight into the independence of the scalar triple product from the point on the line chosen as the reference point of the calculation. The general process (not referenced to Figure 1) to calculate the moment of a force about a specified axis is as follows: The magnitude of a moment about a line segment connecting points P and Q due to a force F applied at point R (with R not on the line through P...
Suppose that a scalar field is constant on a surface As shown in the lectures. there are two methods that one might use to obtain the normal to the surface, and they give the same direction (a) Let r(u, v) be a parametric form for the surface S. Use the vector identity to show that Our ar-λ▽u where λ is a scalar field. [Note: no marks will be awarded for simply stating that a term is zero. If it is...
Only the Matlab part !!! Question 2 For the following vector fields F determine whether or not they are conservative. For the conservative vector fields, construct a potential field f (i.e. a scalar field f with Vf - F) (a) F(z, y)(ryy,) (b) F(z, y)-(e-y, y-z) (c) F(r, y,z) (ry.y -2, 22-) (d) F(x, y, z)=(-, sin(zz),2, y-rsin(x:) Provide both your "by hand" calculations alongside the MATLAB output to show your tests for the whether they are conservative, and to...
(10 points) The work done by a force is the scalar product of the force and displacement vectors, i.e W F x and the power is given by the dot product between the force and the velocity vector, i.e. P F.V . For a force vector, F 2x i+10y j- (x+5y) k and a displacement vector, x=t i+t j+2t k, calculate the work done by the force and the power required. Based on your answer, what can you say about...
Q5. Consider a Lagrangian for some scalar field φ interacting with electric and magnetic fields in the following way: AL where F μν €μυρσ ρσ/2. Obtain the Maxwell equations in vector form, and show that two of them are same as in regular EM and two are different. This theory is supposed to describe ǎ hitherto undetected particle called the "axion", a possible candidate for dark matter (15 pts).
Problem 3 Determine the gradient of the scalar field, and verify with MATLAB
1) (a) The conjugation function on C" is NOT a linear transformation when the scalar field is C, for any positive integer n. However, it IS true when the scalar field is R. Show that the conjugation function T:C" C", where T () = 7 is a linear transformation for the vector space Cover R. (b) Show that CR2m as vector spaces over R.