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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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Knowledge Check

  • Electric field in EM waves is E= E_0sin (kz-omegat)(hati+hatj) , then equation of magnetic field is:

    A
    `B=B_0(-hati+hatj) sin (kZ-omega t)`
    B
    `B=B_c(hati+hatj) sin (kZ-omega t)`
    C
    `B=B_c(-hatj+hatk) sin (kZ-omega t)`
    D
    `B=B_0(hati+hatj) sin (kZ-omega t)`

  • In a region of space the electric field in the x -direction and proportional to x i.e., vec(E )=E_(0)xhat(i) . Consider an imaginary cubical volume of edge a with its sides parallel to the axes of coordinates. The charge inside this volume will be

    A
    zero
    B
    `epsilon_(0)E_(0)a^(3)`
    C
    `(1)/(epsilon_(0))E_(0)a^(3)`
    D
    `(1)/(6)epsilon_(0)E_(0)a^(3)`

  • Uniform electric field exists in a region and is given by vecE = E_0hati + E_0hatj . There are four points A(-a,0), B(0,-a), C(a,0), and D(0,a) in the xy plane. Which of the following is the correct relation for the electirc potential?

    A
    `V_A=V_CgtV_B=V_D`
    B
    `V_A=V_BgtV_C=V_D`
    C
    `V_AgtV_CgtV_B=V_D`
    D
    `V_AltV_CltV_BltV_D`