Negatively-charged balloon X attracts balloon Y and attracts balloon Z. Balloon Z b C d e X must be neutral must be positive cannot determine from this scenario may be negative may be neutral may be positive X Z Select all that apply.

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)

Calculate the flux of the vector field F(x, y, z) = (5x + 8)i through a disk of radius 7 centered at the origin in the yz-plane, oriented in the negative x- direction.

Consider a solid sphere (e.g., a planet) with mass M and radius R. The volume mass density for this planet is given by r2 p(r) = for rR where A is a constant with the units of kg/m'.

Please answer (a), (b), and (c), showing all work.

#6. One the following vector fields F(x, y,z)= xyz*i+x²z*j+4x²yz°k G(x,y,z)= i+ (sin z) j+ y (cos z)k is conservative and other is not. (a) Determine whether F is conservative or not. (b) Determine whether G is conservative or not. (c) For the vector field that is conservative find a functionf such that the vector field is equal to Vf.

consider the parallelepiped with sides: A=3i+2j+k، B=i+j+2k, c=i+3j+3k, then 1-Find the rolume of the paralldepiped 2-Find the area of the face determined by A and B. 3-Find the angle between the vactor C and the plane containing the face determined by A and B

The figure shows a proton p and an electron e that are at rest on a line in the plane of the page. They are between two large uncharged parallel plates (not shown). Assume the particles are far apart so that their mutual gravitational and electrostatic forces are negligible. An electric field directed toward the top of the page is now established between the plates. Which of the following describes the velocity of the center of mass of the particles before either of them hits one of the plates? Toward the top of the page and increasing Toward the top of the page and constant Toward the bottom of the page and increasing Zero

A dipole with charge 6.0 × 10−6 C and separation of 4.0 × 10−3 m is immersed in an E field of 3.0 × 106 N/C. Calculate the maximum torque.

Consider a hollow sphere with radius R and surface potential V(0) = 5 – cos0. Use the general solution of Laplace equation V (r, 8) = E3 (A;r' +)Pi(cos®) to find an expression for the potential at any point outside the sphere. a) V(r,8) : R3 cose R3 cose 2R cose b) V(r, 8) 3R c) V(r,6) R3 R cose 4R 5R d) V (r, 8) - cose e) V(r,8) a b C d e