The electric field at 2R from the centre of a uniformly charged non-co...
Explanation:
Let Q be the total charge of the sphere. Since the sphere is uniformly charged, the charge density is given by ρ = Q/(4/3πR^3).
Electric field at 2R from the centre:
Using Gauss’s law, we can find the electric field at a distance 2R from the centre of the sphere.
∫E.dA = Q/ε₀
Where ε₀ is the permittivity of free space. The electric field is constant over the Gaussian surface, which is a spherical surface of radius 2R.
E.4π(2R)^2 = Q/ε₀
E = Q/(4πε₀R^2)
E = ρ(4/3πR^3)/(4πε₀R^2) = ρR/(3ε₀)
E = (Q/(4/3πR^3))(R/(3ε₀)) = Q/(3πε₀R^2)
E = kQ/R^2
Where k = 1/(4πε₀) is the Coulomb constant.
Therefore, the electric field at a distance 2R from the centre of the sphere is E = kQ/R^2.
Electric field at R/2 from the centre:
Using the principle of superposition, we can find the electric field at a distance R/2 from the centre of the sphere.
Let P be a point at a distance R/2 from the centre of the sphere. We can divide the sphere into concentric shells of thickness dr and radius r, where r ranges from 0 to R.
The electric field due to each shell at point P is given by dE = k(dq)/r^2, where dq is the charge on the shell.
The total charge on the shell of radius r is Q(r) = (4/3)πr^3ρ = (4/3)πr^3(Q/(4/3πR^3)) = (r^3/R^3)Q.
The electric field due to the shell of radius r at point P is dE = k(dq)/r^2 = k(Q(r)/R^3)(r^2)/(R/2)^2 = 2kQ(r)/R^3.
The total electric field at point P is obtained by integrating over all the shells:
E = ∫dE = ∫(2kQ(r)/R^3)dr = (2k/R^3)∫Q(r)rdr = (2k/R^3)Q∫r^4dr/R^4 = (2k/R^3)Q(R^5/5R^4) = (2/5)(kQ/R^2).
Therefore, the electric field at a distance R/2 from the centre of the sphere is E = (2/5)(kQ/R^2) = (2/5)E = 0.4E.