(a) Let
(b) Show that
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Introduction to Electrodynamics
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- 3. Given the following scalar potentials (V), calculate the solution for the gradient of V (VV), and plot the vector arrow representation of this vector field over the given limits. (a) V = 15 + r cos o, for 0 < r < 10, and 0 < $ < 2n. (b) V = 100 + xy, for –10 < x < 10,arrow_forward(0,0,0) and the 5. The point o is the origin of the coordinate system, o = coordinates of b are b = (x,y, z). The electric potential at o is zero, V, = 0. Hence, the electric potential at b is V = – E · dr. You can take any path from o to b. (a) Here is one particular path from o to b. First move on a straight line from o to a = : (x,0,0), then from a to a' = (x, Y, 0), and finally from a' to b = (x, y, z). Make a plot of this path, indicating the coordinate system and the locations of a, a' and b. – SE • dr depends only on the y However, the line integral V coordinate of b = (x, y, z). Hence, in the following, we will focus on b = (0, y, 0) and take the straight path from o to b. (b) Compute E · dr for y d result, use this potential next.arrow_forwardA sphere of radius a has potential (sin 2θ)( cos ϕ) on its surface. Find the potential at all points outside the sphere.arrow_forward
- If V (x, y) is the electric potential at a point (x, y) of the xy plane, then the level curves of V are called equipotential curves, so that at all points on this curve the electric potential is the same. Find the range of V and the level curves of the following function:arrow_forwardWhen two charges of the same sign and magnitude are at the adjacent vertices of a square, the electric potential at the center of the square is V. What is the electric force on the charge of the same magnitude and sign positioned at the center of the opposite side, at point P, in addition to the two original charges? Draw the electric force vectors and express your answer in term of V and k. □arrow_forwardQ3/ a) given that the potential v = (2x²y + 2xy²z²)mv 1) Find E at (1,2,3) 2) Py c/m³ b) point p and Q are located at P(1,2,3) and Q(3,2,1) find the vector parallel to PQ with magnitude of 5cm.arrow_forward
- Consider the vector field F(x,y)=(4x^3y^2−2xy^3)i+(2x^4y−3x^2y^2+4y^3) a. Find the potential function f(x,y). b. Determine the work done by the vector field in moving a particle along a curve C from (1,2) to (3,1).arrow_forwardConsider the following potential V(x) = 3x2 + 4.x + 3. The equilibrium and it is a position is located at equilibriumarrow_forwardUsing the general expression V(r,0) E (A,r BỊ + 1) P(cose) for the potential at a distance r from the center of a sphere shell of radius R. The acceptable expressions for potentials inside and outside the shell are: V (r,0) : E, Ar' P,(cose) (rarrow_forward247. Consider a particle moving in the region x> 0 under the influence of the potential a + а, U(x) %3D where U, = 1 J and a = 2 m. Plot the potential, find the equilibrium points, and determine whether they are maxima or minima.arrow_forwardConsider two points in an electric field. The potential at point 1, V1, is 29 V. The potential at point 2, V2, is 173 V. An electron at rest at point 1 is accelerated by the electric field to point 2. Write an equation for the change of electric potential energy ΔU of the electron in terms of the symbols given. Find the numerical value of the change of the electric potential energy in electron volts (eV). Express v2, the speed of the electron at point 2, in terms of ΔU, and the mass of the electron me.arrow_forwardConsider the vector field T(x) = y³î – x³j. Which of the following expressions is a vector potential for v(x) ? Select one: a. there is no vector potential since V x u + 0 b. There is no vector potential since V · U = 0 %D c. 3(x2 + y²)k d. –3(x? + y²)k e. none of the other answers is correct. O f.-zx'j + x*k 1 4arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning