Solution

With

ω = B / m

, magnetic force does no work, so

∣ \overset{x}{˙} ∣

is constant; its zero

z

component also makes

\overset{z}{˙}

constant. For the stated data,

\overset{x}{˙} = u cos ω t, \overset{y}{˙} = - u sin ω t,

x = \frac{u}{ω} sin ω t, y = \frac{u}{ω} (cos ω t - 1) .

This is a circle of radius

u / ω

centred at

(0, - u / ω)

With

α = μ / m

, the complex velocity is

\overset{x}{˙} + i \overset{y}{˙} = u e^{- (α + iω) t}

. Hence

x = \frac{u [ α ( 1 - e ^{- α t} c o s ω t ) + ω e ^{- α t} s i n ω t ]}{α ^{2} + ω ^{2}},

y = \frac{u [ α e ^{- α t} s i n ω t - ω ( 1 - e ^{- α t} c o s ω t )]}{α ^{2} + ω ^{2}} .

The path spirals into

u (α, - ω) / (α^{2} + ω^{2})

. As

u

varies, this lies on

y = - (ω / α) x

, at distance

\frac{u}{α ^{2} + ω ^{2}} = \frac{m u}{μ ^{2} + B ^{2}}

from the origin.

Solved by gpt-5.6-sol high.

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