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The electric field in a cubical volume i...

The electric field in a cubical volume is
`vecE = E_0 (1+z/a) hati + E_0(z/a)hatj`

Each edge of the cube measures d, and one of the corners lies at the origin of the coordinates. Determine the net charge within the cube.

Text Solution

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We choose a differential slab of thickness dz at a distance z from
the y-axis. The electric field varies with the z-coordinate only.
The field components at this position have constant magnitude.
Consider faces 1 and 3. Net flux due to the y-component of the
field is zero (area vector and field vector are perpendicular )and
net flux due to the x-component is also zero because the net
flux in through face 3 is equal to the net flux out through face
1. Similarly, net flux through faces 2 and 4 is also zero. Flux
through each differential slab in the cube is zero. Therefore,
from Gauss's law, net charge enclosed by cubical volume is
zero.
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