A cube, centered on the origin, with sides that are parallel to the x-y, y-z, and x-z planes

Question

A cube, centered on the origin, with sides that are parallel to the x-y, y-z, and x-z planes and have an edge length of L, is immersed in an electric field given by E(x) = E₀(αx³i + 2j + 5k). Refer to the figure.

a.  Integrate to find an expression for the total electric flux through the cube in terms of defined quantities and enter the expression. 

b. If L = 7.4 m, α = 1.2 m^-3, and E₀ = 28 V/m , what is the value of the flux, in units of volt•meter? 

c. If the electric field is E(x) = E₀(βx²i + 2j + 15k) with β = 2.5, what is the value of the flux, in units of volt•meter? 

Answer 1

a. Use the Gauss divergence theorem where flux =integral of Edot n dS and becomes the integral of the divergence of E dV. Using the given electric field in a the divergence is 3E₀ax². Integrate this between -L/2 to L/2. Flux becomes 3E₀a (x³/3)_-L//2^L/2y_-L/2^L/2z_-L/2^L/2. This gives E₀aL⁵/4.

b. Plug in the values to find the numerical value of the flux.

c. Solve this step the same way as a.