The Electric Field due to an Unknown Charge Distribution

The electric field due to an unknown charge distribution is given by the equation E = q/R^2 * exp(-4r), where E is the electric field, q is the charge, R is the distance from the charge, and r is the distance from the charge normalized by R.

Explanation:

To find the total integrated charge over all space, we need to integrate the charge density over all space. The charge density is given by the equation dQ = ρ dV, where dQ is the charge contained within a small volume element dV and ρ is the charge density.

Integration of the Charge Density:

To integrate the charge density over all space, we need to determine the limits of integration. Since the charge distribution is unknown, we will assume that it extends to infinity in all directions. Therefore, the limits of integration will be -∞ to +∞ for all three spatial dimensions.

The charge density ρ can be expressed in terms of the electric field E as ρ = ∇ · E, where ∇ is the divergence operator. Taking the divergence of the electric field equation, we have:

∇ · E = (∇ · (q/R^2 * exp(-4r)))
= q/R^2 * (∇ · exp(-4r))

The divergence of exp(-4r) can be calculated using the chain rule:

(∇ · exp(-4r)) = (∂/∂x + ∂/∂y + ∂/∂z) · exp(-4r)
= -4(exp(-4r)) · (∂r/∂x + ∂r/∂y + ∂r/∂z)

Since r is the distance from the charge normalized by R, we have r = √(x^2 + y^2 + z^2)/R. Taking the partial derivatives, we get:

∂r/∂x = x/(R√(x^2 + y^2 + z^2))
∂r/∂y = y/(R√(x^2 + y^2 + z^2))
∂r/∂z = z/(R√(x^2 + y^2 + z^2))

Substituting these values into the expression for (∇ · exp(-4r)), we get:

(∇ · exp(-4r)) = -4(exp(-4r)) · (x/(R√(x^2 + y^2 + z^2)) + y/(R√(x^2 + y^2 + z^2)) + z/(R√(x^2 + y^2 + z^2)))
= -4(exp(-4r)) · (x + y + z)/(R√(x^2 + y^2 + z^2))

Now, substituting this expression for (∇ · exp(-4r)) back into the equation for ρ, we have:

ρ = q/R^2 * (∇ · exp(-4r))
= q/R^2 * (-4(exp(-4r)) · (x + y + z)/(R√(x^2 + y^2 + z^2)))

Integration Limits: