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A region is specified by the potential f...

A region is specified by the potential function `V = 2x^(2) + 3y^(3) - 5z^(2)`. Calculate the electric field intensity at a point (2,4,5) in this region.

Text Solution

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We know , `vec(E ) = E_(x)hat(i)+E_(y)hat(j)+E_(z)hat(k) = -(dV)/(dx)hat(i)-(dV)/(dy)hat(j)-(dV)/(dz)hat(k)`
Now, `V = 2x^(2)+3y^(3)-5z^(2)`
`:. (dV)/(dx) = 4x , (dV)/(dy) = 9y^(2),(dV)/(dz) = -10z`
[here, x,y,x are mutually independent variable]
At the point (2,4,5),
`(dV)/(dx) = 4 xx 2 = 8, (dV)/(dy) = 9 xx (4)^(2) = 144`,
`(dV)/(dz) = -10 xx 5 = -50`
`:.` At the point `(2,4,5), vec(E ) = -8hat(i) - 144hat(j) + 50hat(k)`
`:. E = sqrt(E_(x)^(2)+E_(y)^(2)+E_(z)^(2))`
`= sqrt((8)^(2)+(144)^(2)+(50)^(2)) = 152.64` units.
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