VECTOR CALCULUS
10.3 The general equation of motion of a (non-relativistic) particle of massmand
chargeqwhen it is placed in a region where there is a magnetic fieldBand an
electric fieldEis
m ̈r=q(E+ ̇r×B);
hereris the position of the particle at timetand ̇r=dr/dt, etc. Write this as
three separate equations in terms of the Cartesian components of the vectors
involved.
For the simple case of crossed uniform fieldsE=Ei,B=Bj,inwhichthe
particle starts from the origin att=0with ̇r=v 0 k, find the equations of motion
and show the following:
(a) ifv 0 =E/Bthen the particle continues its initial motion;
(b) ifv 0 = 0 then the particle follows the space curve given in terms of the
parameterξby
x=mE
B^2 q(1−cosξ),y=0,z=mE
B^2 q(ξ−sinξ).Interpret this curve geometrically and relateξtot. Show that the total
distance travelled by the particle after timetis given by
2 E
B∫t0∣∣
∣
∣sinBqt′
2 m∣∣
∣
∣dt′.
10.4 Use vector methods to find the maximum angle to the horizontal at which a stone
may be thrown so as to ensure that it is always moving away from the thrower.
10.5 If two systems of coordinates with a common originOare rotating with respect
to each other, the measured accelerations differ in the two systems. Denoting
byrandr′position vectors in framesOXY ZandOX′Y′Z′, respectively, the
connection between the two is
̈r′= ̈r+ω ̇×r+2ω× ̇r+ω×(ω×r),
whereωis the angular velocity vector of the rotation ofOXY Zwith respect to
OX′Y′Z′(taken as fixed). The third term on the RHS is known as the Coriolis
acceleration, whilst the final term gives rise to a centrifugal force.
Consider the application of this result to the firing of a shell of massmfrom
a stationary ship on the steadily rotating earth, working to the first order in
ω(= 7. 3 × 10 −^5 rad s−^1 ). If the shell is fired with velocityvat timet=0andonly
reaches a height that is small compared with the radius of the earth, show that
its acceleration, as recorded on the ship, is given approximately by
̈r=g− 2 ω×(v+gt),
wheremgis the weight of the shell measured on the ship’s deck.
The shell is fired at another stationary ship (a distancesaway) andvis such
that the shell would have hit its target had there been no Coriolis effect.
(a) Show that without the Coriolis effect the time of flight of the shell would
have beenτ=− 2 g·v/g^2.
(b) Show further that when the shell actually hits the sea it is off-target by
approximately
2 τ
g^2
[(g×ω)·v](gτ+v)−(ω×v)τ^2 −1
3
(ω×g)τ^3.(c) Estimate the order of magnitude ∆ of this miss for a shell for which the
initial speedvis 300 m s−^1 , firing close to its maximum range (vmakes an
angle ofπ/4 with the vertical) in a northerly direction, whilst the ship is
stationed at latitude 45◦North.