Chapter 6.1, Problem 6.2P

Assume MKS units... Let Q be an open subset of R³. Let B: :Q - R³ be a continuous vector .field, representing a magnetic field in 3-D space. 7 Let P be a particle with charge q E R and mass m > 0. If p is at position (x. y, z) in Q and R³ is the velocity of p, at time t, then p feels a force 7(7,7) given by - 7(7,J) := q V × B (7) . Suppose that p moves along a curve C as time t varies from a to b, and that p has position vector (t) and instantaneous velocity (t) at time t. ř (1) Explain why the two vectors 7'(t) × È(7(t)) and 7'(t) are perpen- dicular at every time t = [a, b]. (2) Using Part (1), calculate W := the work done on the particle p by the force as p moves from D = 7(a) to E = √ (b) along C. F (3) Prove that ((t)||²)=27' (t) • F(t), at each time t. (4) Using Parts (2) and (3), and Newton's Second Law, prove that if the magnetic force - ₹(7,7) is the total force on p at every time t, then p moves along C at a constant speed. dt

A very long, straight current-carrying wire is bent at the middle two different ways, sothat it takes the shapes depicted in figures a) and b). Find the direction and magnitude of themagnetic field in both cases at the center C of the semicircle (of radius R = 10 cm), if the currentflowing in the wire is I = 2 A.

A charge q = 54 C is located at (9, 8) and we want to find the electric field at pint p (3, 5). Find the source to point vector. Use the following constants if necessary. Coulomb constant, k = 8.987 × 10° N · m² /C² . Vacuum permitivity, €o = 8.854 × 10-12 F/m. Magnetic Permeability of vacuum, µo = 12.566370614356 ×x 10-7 H/m. Magnitude of the Charge of one electron, e = -1.60217662 × 10¬19 C. Mass of one 9.10938356 x 10-31 kg. Unless specified otherwise, each symbol carries their usual meaning. For example, µC means micro coulomb electron, me x component of the vector Give your answer up to at least three significance digits. y component of the vector Give your answer up to at least three significance digits.