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If current density vecJ is defined as ...

If current density `vecJ` is defined as a vecto with magniutde equal tpo current per unit area, area being normal to the current and direction in which current flows, show that `I= int vecJ.dvecs`

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By definition
`vecJ = (dI)/(ds cos theta ) hatk`
or ` hatk dI = vecJ ds cos thetat `
or ` hatk dI = J ds cos theat hatk" " [as hatJ = J hatk]`
Now , taking scalar product of both sides with `hatk`,
`hatk .hatk dI = J ds cos theta hatk. hatk`
or `dI =J ds cos theta` " "[as `hatk . hatk = 1]" "....(i)`
But by denfinition of scalar product of two vectors,
`vecJ. dvecs = J ds cos theta" " ......(ii)`
So, from Eqns. (i) and (ii), we have
`dI = vecJ . dvecs or I = int vecJ.dvecs`
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