PHY5200 F06

Chapter 2: Projectiles and Particles

Reading

Taylor 2.6-2.7 (today) and 3.1-3.2 (Friday)

Recall

The equation of motion for a charge q in magnetic field B is

v_x[dot] = ωv_y

v_y[dot] = -ωv_x

where ω=qB/m. By writing η=vx+ivy, we transformed the two coupled differential equations for the motion of a charge in magnetic field to a single differential equation of a complex variable

η[dot] = -iωη

with solution

η = Ae^-iωt.

Let's review complex numbers and learn what the exponential of a complex number means.

Complex Exponentials

The complex number i=(-1)^½, so obviously i² = -1. An arbitrary complex number is written z = x + iy and can be represented in the complex plane, a two dimensional coordinate system where the real part of the number, x, is plotted along the horizontal (or x) axis and the imaginary part, y, is plotted along the vertical (or y) axis. Complex numbers can be added, subtracted, multiplied, and divided, as long as care is taken to keep track of the complex as well as the real part. Complex numbers resemble two-dimensional vectors in this sense.

Recall that the exponential function is equivalent to the infinite series

e^z = 1 + z + z²/2! + z³/3! + ... = ∑_n=0^∞ zⁿ/n!

and the derivative is

(d/dz)(Ae^kz) = k(Ae^kz).

These relations hold for complex numbers as well as real numbers.

But what does it mean to exponentiate a complex power? To investigate, it is useful to look at the simplest case of a complex exponential

e^iθ = 1 + iθ + (iθ)²/2! + (iθ)³/3! + ...

This expression can be simplified using i²=-1, and grouping all even powers of i into a real part, and all odd powers into an imaginary part:

e^iθ = [1 - θ²/2! + θ⁴/4! -...] + i[θ - θ³/3! + ...].

These two parts should look familiar: the real piece is the Taylor series for cosθ while the imaginary piece is isinθ

e^iθ = cosθ + isinθ.

This expression is known as Euler's formula. It is extremely useful in physics, and can be helpful to quickly (re)derive trigonometric relations such as for sine or cosine of a sum of angles.

Using the complex exponential, we can write the general solution as (even if A is a complex number)

η = Ae^-iωt = ae^iδe^-iωt = ae^i(δ-ωt)

Solution for the Charge in a B Field

With the solution for the velocity as a function of time, we can now solve for the position. To do that, it is convenient to define another complex variable, ξ, pronounced xi,

ξ = x + iy.

The derivative of ξ is η

ξ[dot] = x[dot] + iy[dot] = v_x + iv_y = η

so we can solve for ξ, and hence the positions x and y, by integrating η

ξ = ∫ ηdt = ∫ Ae^-iωtdt = (iA/ω)e^-iωt + constant.

Since x is the real part of ξ and y is the imaginary part, we can write, dropping the constant since it basically corresponds to a displacement of the motion:

x+iy = Ce^-iωt

When t=0, the exponential is 1, so C corresponds to the position of the particle

C = x₀ + iy₀

r = v/ω = (mv)/(qB) = p/(qB)