Question 1: Given the following vector fields, deduce which fields can possibly represent an electrostatic or magnetostatic field or none of them in free space. a) Ä=y°zâ, + 2(x+1)yzâ, -(x+1)z² â. z+1 b) B = |cospâ, + â, (in cylindrical coordinates)
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- Consider the vector field ʊ(r) = (x² + y²)êx + (x² + y²)êy + z²êz. Decompose the vector field (r) into the sum of two other vector fields, a (r) and 5(r), such that a(r) has no divergence (it is solenoidal) and 5 (r) has no curl (it is irrotational). The answer is not unique. This is the Helmholtz decomposition.Evaluate the line integral, where C is the given curve. Sc (x + yz)dx + 2x dy + xyz dz C consists of line segments from (1, 0, 1) to (2, 2, 1) and from (2, 2, 1) to (2, 4, 3). The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = is F(r) = Kr/1r|³ where K is a constant. Find the work done as the particle moves along a straight line from (5, 0, 0) to (5, 1, 5). Find the mass and center of mass of a wire in the shape of the helix x=t, y = 5cos(t), z = 5sin(t), 0 st ≤ 2π, if the density at any point is equal to the square of the distance from the origin. (mass) ) (center of mass)An electron is projected with an initial speed v0 = 1.70×106 m/s into the uniform field between the parallel plates in (Figure 1). Assume that the field between the plates is uniform and directed vertically downward, and that the field outside the plates is zero. The electron enters the field at a point midway between the plates. If the electron just misses the upper plate as it emerges from the field, find the magnitude of the electric field. Express your answer in newtons per coulomb. Suppose that in the figure the electron is replaced by a proton with the same initial speed v0v0. Would the proton hit one of the plates? What would be the direction of proton's displacement? displacement is upward displacement is downward
- A disc, shaped like a quarter circular arc with inner radius a and outer radius b, carries a surface charge density given by σ(θ) = αcosθ where θ is the angle to the x-axis in cylindrical coordinates and α a positive constant. Determine the total charge carried by the diskWe have calculated the electric field due to a uniformly charged disk of radius R, along its axis. Note that the final result does not contain the integration variable r: R. Q/A 2€0 Edisk (x² +R*)* Edisk perpendicular to the center of the disk Uniform Q over area A (A=RR²) Show that at a perpendicular distance R from the center of a uniformly negatively charged disk of CA and is directed toward the disk: Q/A radius R, the electric field is 0.3- 2€0 4.4.1bThe figure below shows four small charged spheres arranged at the corners of a square with side d = 25.0 cm. (Let q₁ = +4.00 nC, 92 = +6.00 nC, 93 = +9.00 nC, and 94 = +5.00 nC. Assume 93 is located at the origin and +x axis is to the right and the +y axis is up along the page. Express your answers in vector form.) 91 d 92 93 d 94 (a) What is the electric field at the location of the sphere with charge +9.00 nC? N/C (b) What is the total electric force exerted on the sphere with charge +9.00 nC by the other three spheres? = N d d
- An electron travels from right to left at 3 x 10° m/s when it encounters the B field pictured below. The magnitude of the field is 1Tesla. Draw a picture of the electron's path in the field with the correct direction. Calculate the radius of curvature. х хAn electron is circularly orbiting a proton. The magnitude of acceleration of the electron is 1.44×10–8 m/s2 . What is the electron’s orbital radius?Suppose a unidirectional vector field E exists across an enclosed surface A as shown below. which among the following is/are true? 153 E The divergence volume integral value will only be equal to 0 if E is a uniform field. O The divergence volume integral value will be positive if the magnitude of E is increasing along the positive z-hat direction. The divergence volume integral value will always be equal to 0. Only the top and bottom cylindrical phases have non-zero divergence surface integral values.
- There are three static charges (Q1 = 4 nC, Q2 = - 4 nCn and Q3 = - 4nC) located in different locations of Fig. 1. Calculate the net electric field at the given point P.A uniform electric field of magnitude 43.1N/C is parallel to the x axis. A circular loop of radius 25.7cm is centered at the origin with the normal to the loop pointing 27.4 above the x axis. A). Calculate the electric flux in, newton squared meters per coulomb, through the loop. B). To what angle, in degrees from the positive x axis, should the normal of the loop be rotated so that the flux through the loop becomes 0.419N⋅m2/C?Problem 3: UP 6.53 Charge is distributed uniformly with a density p throughout an infinitely long cylindrical volume of radius R. Show that the field of this charge distribution is directed radially with respect to the cylinder and that E(s) = ps 2€0 PR² 2€ S S≤R SZR