Calculate the electric flux piercing a cube?

[vmr101]

Homework Statement

Consider four point charges q1, q2, q3 and q4, located at r1, r2, r3 and r4, respectively.
(a) Calculate the electric flux piercing a cube (with side a and centered at r0 = (0, 0, 0) that contains all of these charges.
(b) Calculate the electric field of the four charges as the function of r.
(c) Calculate the Coulomb forces acting on all the four charges.
(bonus) Calculate the divergence of the electric field created by these charges.

Homework Equations

Nothing is given.
Using k = 1/4∏ε

The Attempt at a Solution

a) Electric Flux ∅ = ∫E dA
Each of the size sides receive the same flux as each other, therefore one side will receive 1/6 of the flux ∅(a) = 1/6 ∫E dA

b) Due to the superposition principal E = ƩE = E1 + E2 + E3 + E4
so E = ƩE = k Ʃ q(i)/r(i)^2
E = k (q(1)/r(1)^2 + q(2)/r(2)^2 +q(3)/r(3)^2 +q(4)/r(4)^2)

c) Due to the superposition principal
F = ƩF = F1 + F2 + F3 + F4
F = kq Ʃ q(i) * (r -r(i)) / |(r - r(i))|^3

I am not sure if I am on the right track as there is not much given. We have been learning Gauss;s laws & Maxwells equations. Thanks for any guidance.

 

[PhysicsGente]

I have not checked your answers but I would too use the same reasoning. It seems to me you are in the right track.

 

[vmr101]

Anyone else have any feedback on this?

 

[bayan]

For part a) it says to find the flux piercing a cube, if cube encloses all 4 charges then you don't need integration.

I am also doing a similar assignment and for part c) I think they want us to find the forces that the other 3 charges apply to one. i.e force that charge 2,3,4 exert on 1 and 1,3,4 exert on 2 etc...

 

[Nickie]

I would first clarify any uncertainties with the given problem. For example, are the charges all positive or can they be a mixture of positive and negative? Are the charges stationary or are they in motion? These details can affect the calculation of electric flux and electric field.

Assuming all the charges are positive and stationary, here is my attempt at a solution:

a) To calculate the electric flux piercing a cube, we can use Gauss's law, which states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (ε0). In this case, the cube is our closed surface and the charge enclosed is the sum of all four point charges. So, the electric flux piercing the cube is given by:

Φ = q1 + q2 + q3 + q4 / ε0

b) To calculate the electric field at a point r due to the four charges, we can use the superposition principle, which states that the total electric field at a point is the vector sum of the individual electric fields due to each charge. So, the electric field at point r is given by:

E = k(q1/r1^2 + q2/r2^2 + q3/r3^2 + q4/r4^2)

where k is the Coulomb constant and r1, r2, r3, and r4 are the distances from each charge to point r.

c) To calculate the Coulomb forces acting on each charge, we can use Coulomb's law, which states that the force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. So, the force on q1 due to the other three charges is given by:

F1 = kq1(q2/r12^2 + q3/r13^2 + q4/r14^2)

Similarly, the forces on q2, q3, and q4 can be calculated using the same formula.

Bonus) To calculate the divergence of the electric field created by the four charges, we can use the divergence theorem, which states that the flux through a closed surface is equal to the volume integral of the divergence of the vector field within that surface. So, the divergence of the electric field is given by:

∇·E = ρ / ε0

where ρ is the charge density, which in this case is the