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.7Let 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 aforce 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 thatp 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 theforce 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 themagnetic force - ₹(7,7) is the total force on p at every time t, then pmoves along C at a constant speed.dt

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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 spacc. Let p be a particle with charge q ER and mass m > 0. If p is at position 7=(x. y, z) in Q and E R³ is the velocity of P, at time t, then р feels a force ,) given by F(7₁7)=qx 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. (1) Explain why the two vectors (t) × (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. (3) Prove that d (|| T (t)||²³) = 2 T '(t) • F(t), dt 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.

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 spacc. Let p be a particle with charge q ER and mass m > 0. If p is at position 7=(x. y, z) in Q and E R³ is the velocity of P, at time t, then р feels a force ,) given by F(7₁7)=qx 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. (1) Explain why the two vectors (t) × (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. (3) Prove that d (|| T (t)||²³) = 2 T '(t) • F(t), dt 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.