What is the net electric flux through the box?

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A 1.0 cm x 1.0 cm x 1.0 cm box with its edges aligned with the xyz axes is in the electric field E=<350x+150>i N/C, where x is in meters. What is the net electric flux through the box.

I’m lost. :/

2 Answers

  • The question is badly worded. You can’t have ‘net … flux through the box’. You can only have net flux into (or out of) the box.

    Area of each face A = 0.01m x 0.01m = 1×10⁻⁴m²

    For convenience, take the +x (i) direction to be to the right.

    If the left face is at x = X, then the right face is at x = X+0.01 (metres).

    The electric field has only an x-component, so flux passes through only the left and right faces.

    Flux passing through left face is Φₗ = EA = (350X+150)*0.0001

    This is in the +x direction so it enters the box.

    Flux passing through right face is Φᵣ = EA = (350(X+0.01)+150)*10⁻⁴

    This is also in the +x direction so it leaves the box.

    The flux leaving is greater than the flux entering, so:

    Net flux, Φnet, *leaving* box is

    Φnet = Φᵣ – Φₗ

    . . . . .= (350(X+0.01)+150)*10⁻⁴ – (350X+150)*10⁻⁴

    When you evaluate this, most terms cancel leaving:

    Φnet = 350*0.01*10⁻⁴

    . . . . .= 3.5×10⁻⁴ Vm (the unit of electric flux is volt metre).

    (If you use the convention that flux entering the box is positive, then the answer would be -3.5×10⁻⁴ Vm.)

  • Two ways of doing it here:

    The field is in the x-direction only, so the flux is nonzero through the squares of the cube at x=0 and x=1.

    At x=0 the field is a constant 150 N/C, and over the area of 1 cm^2 the flux therefore is (negative, into the oriented surface) -150 N cm^2/C

    At x=1 the field is a constant 500N/c, and so the flux through that surface is 500 N cm^2/C .

    The total flux therefore is 350 N cm^2/C, or 3.50*10^-4 N m^2 / C.

    Second method: the flux integral equals the volume integral of the divergence. The divergence is ∂Ex / ∂x = 350 N C / m, and integrating this constant over the volume (10^-6 m^3) gives 3.50*10^-4 N m^2/C

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