A spherical cloud of charge of radius R contains a total charge +Q with a nonuniform volume charge density that varies according to the equation. p(r) =P0(1-r/R) for r is less than or equal to R and p = 0 for r>R. Determine the magnitude E of the elcetric field. Express your answer in terms of Q, R and fundamental contants. p and p0 looks like a greak letter i just use p instead.
In this case, because you are beyond the surface of the sphere you should be able to treat it like a point charge.
To determine the magnitude of the electric field (E) created by a spherical cloud of charge with a nonuniform volume charge density, we can use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (ε₀).
1. Consider a Gaussian surface in the form of a sphere of radius r (where r ≤ R) centered at the center of the cloud of charge.
2. The electric field (E) and the charge density (ρ) are radial, and since the cloud of charge is spherically symmetrical, the electric field magnitude will only depend on the distance from the center (r).
3. The electric flux (Φ) passing through the Gaussian sphere is equal to the product of the magnitude of the electric field (E) and the surface area of the sphere (A):
Φ = E * A = E * 4πr²
4. The charge enclosed within the Gaussian sphere can be calculated by integrating the volume charge density (ρ) over the volume of the sphere:
Qenclosed = ∫ρ dV
Qenclosed = ∫(P₀(1 - r/R)) * (4/3)πr³ dr (from 0 to r)
5. Simplifying the integral, we get:
Qenclosed = (4/3)P₀π * ∫(r³ - (r⁴/R)) dr (from 0 to r)
6. Integrate each term separately:
Qenclosed = (4/3)P₀π * [(r⁴/4) - (r⁵/5R)] (from 0 to r)
Qenclosed = (P₀π/3) * (r⁴ - (4/5) * (r⁵/R)) (from 0 to r)
7. Now, applying Gauss's Law:
Φ = E * 4πr² = (Qenclosed / ε₀)
E * 4πr² = [(P₀π/3) * (r⁴ - (4/5) * (r⁵/R))] / ε₀
8. Rearrange the equation to solve for E:
E = [[(P₀π/3) * (r⁴ - (4/5) * (r⁵/R))] / (4πr²ε₀)]
9. Simplify the expression:
E = (P₀ / (3ε₀)) * (r² - (4/5) * (r³/R))
10. Finally, since we want to find the magnitude of the electric field, we can disregard the negative sign and substitute r with R:
E = (P₀ / (3ε₀)) * [(R² - (4/5) * (R³/R))]
E = (P₀ / (3ε₀)) * [(R² - (4/5) * R²)]
E = (P₀ / (3ε₀)) * [(R² * (1 - (4/5)))]
E = (P₀ / (3ε₀)) * [(R² / 5)]
E = (P₀R²) / (15ε₀)
Therefore, the magnitude of the electric field (E) is (P₀R²) / (15ε₀), where P₀ represents the volume charge density, R is the radius of the spherical cloud of charge, and ε₀ is the permittivity of free space.
To determine the magnitude of the electric field (E) generated by the spherical cloud of charge, we can use Gauss's Law. Gauss's Law states that the electric field at a point outside a closed surface is proportional to the total charge enclosed by that surface.
Let's consider a spherical Gaussian surface of radius r, centered at the center of the charge cloud.
For r > R (outside the cloud), the charge enclosed by the Gaussian surface is zero since the charge density is zero beyond the radius R. Hence, the electric field outside the cloud is also zero.
For r ≤ R (inside the cloud), the charge enclosed by the Gaussian surface is the total charge +Q. We can write this as:
Q_enclosed = (4/3)πr^3 * p(r)
Given p(r) = P0(1 - r/R), we can substitute this into the equation:
Q_enclosed = (4/3)πr^3 * P0(1 - r/R)
Now, applying Gauss's Law, we have:
∫E · dA = Q_enclosed / ε0
Here, ∫E · dA represents the flux of the electric field through the Gaussian surface, and ε0 is the permittivity of free space.
Using symmetry arguments (since the charge cloud is spherically symmetric), we can conclude that the electric field magnitude (E) is constant over the Gaussian surface. Thus, we can take E outside of the integral:
E * ∫dA = Q_enclosed / ε0
The integral of dA over the spherical surface is simply 4πr^2 (the surface area of a sphere with radius r). Applying this, we get:
E * 4πr^2 = (4/3)πr^3 * P0(1 - r/R) / ε0
Simplifying, we find:
E = P0Q / (3ε0) * (1 - r/R) / r^2
Therefore, the magnitude of the electric field E, as a function of r, Q, R, P0, and ε0, is given by:
E = P0Q / (3ε0) * (1 - r/R) / r^2