65. Although we usually write Newton's second law for one-dimensional motion in the form F =ma, which holds when mass is constant, d(mv) a more fundamental version is F = . Consider an object dt whose mass is changing, and use the product rule for derivatives to show that Newton's law then takes the form F dm = ma + v dt

If a proton is located on the x-axis in some coordinate system at x0 = -3.2 x 10-5 meters, what is the x-component of the Electric Field due to this proton at a position x = +3.2 x 10-5 meters and on the x axis as the y-axis is 0 giving a number of Newtons/Coulomb?

Consider a single square loop of wire of area A carrying a current I in a uniform magnetic field of strength B. The field is pointing directly up the page in the plane of the page. The loop is oriented so that the plane of the loop is perpendicular to the plane of the page (this means that the normal vector for the loop is always in the plane of the page!). In the illustrations below the magnetic field is shown in red and the current through the current loop is shown in blue. The loop starts out in orientation (i) and rotates clockwise, through orientations (ii) through (viii) before returning to (i). (i) Ø I N - - I N - (iii) (iv) (v) (vii) (viii) a) [3 points] For each of the eight configurations, draw in the magnetic dipole moment vector μ of the current loop and indicate whether the torque on the dipole due to the magnetic field is clockwise (CW), counterclockwise (CCW), or zero. In which two orientations will the loop experience the maximum magnitude of torque? [Hint: Use the…