ADVANCED DYNAMICS — PHY 4241/5227

HOME AND CLASS WORK – SET 8

Download the Special Relativity and Maxwell Equations script and read up to section 1.1.4 (included). Due March 25.

(23) A particle of mass m and electric charge q moves under the influence of a constant magnetic field of the form B(r) = B_{0}ẑ. Obtain the most general solution for the velocity v(t) using Newton’s second law of motion in combination with the Lorentz force

F = mv̇ = \frac{q}{c}v × B .

Homework, due March 28 before class (10 points).

(24) SI units (due March 28 before class, 4 points).

1. How is one second [s] defined?

2. How is one meter [m] defined?

3. How is one kilogram [kg] defined?

4. How is one Newton [N] defined?

5. How is one Ampere [A] defined?

6. How is one Coulomb [C] defined?

7. From the Biot-Savart law it follows that the force exerted by one (infinitely long) wrire on a section of length l of a parallel wire is given by

F = \frac{μ_{0}}{2π} \frac{I_{1} I_{2}}{d} l ,

where μ_{0} is the permeability of vacuum. What is the (exact) value of μ_{0}?

8. Using the above units, Coulomb’s law read

F (r) = \frac{κ_{0} Q_{1} Q_{2}}{r^{2}} , κ_{0} = \frac{1}{4π ϵ_{0}}

where κ_{0} is called Coulomb constant and ϵ_{0} permittivity of vacuum. They can, for instance, be estimated by force measurements. In SI units ϵ_{0} = 8.85418782 × 10^{−12}. What are the physical dimensions of ϵ_{0}?

(25) Consider the 2D rotation

( \begin{matrix}x^{′ 1} \\ x^{′ 4} \end{matrix} ) = ( \begin{matrix}cos(ϕ) & sin(ϕ) \\ − sin(ϕ) & cos(ϕ) \end{matrix} ) ( \begin{matrix}x^{1} \\ x^{4} \end{matrix} )

and substitute ϕ = i ζ, x^{4} = i x^{0}, x^{′ 4} = i x^{′ 0}. Write out the equations for x^{′ 1} and x^{′ 0}. Due in class (4 points).