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An infinitely long thin wire carrying a ...

An infinitely long thin wire carrying a uniform linear static charge density `lambda` is placed along the z - axis. The wire is set into motion along its length with a uniform velocity `V=v hat(k)_(z)`. Calculate the pointing vector `S=(1)/(mu_(0))(vec(E )xx vec(B))`.

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Electric field produced due to infinitely long charged wire,
`vec(E )=(lambda)/(2pi in_(0)a)hat(j) " "` …(1)

a = radius of cylinderical Gaussian surface around wire.
Magnetic field at .a. distance from current carrying conductor,
`vec(B)=(mu_(0)I)/(2pi a)hat(i)`
but `I=(q)/(t)=(lambda L)/(t)=lambda v [because Q=lambda L " and " (L)/(t)=v]`
Here L = length
`therefore vec(B)=(mu_(0)lambda v)/(2pi a) " "` ...(2)
Now pointing vector,
`S=(1)/(mu_(0))(vec(E )xx vec(B))`
`therefore S=(1)/(mu_(0))[(lambda)/(2pi a)hat(j)xx(mu_(0)lambda v)/(2pi a)hat(i)]`
`=(1)/(mu_(0))((lambda)/(2pi a)xx(mu_(0)lambda v)/(2pi a))(hat(j)xx hat(i))`
`=(lambda^(2)v)/(4pi^(2)in_(0)a^(2))(-hat(k)) " "[because hat(j)xx hat(i)=-hat(k)]`
`therefore S=-(lambda^(2)v)/(4pi^(2)in_(0)a^(2))hat(k)`
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