In all spherically symmetrical cases, the electric field at any point must be radially directed, because the charge and, hence, the field must be invariant under rotation. If the density depends on θ θ or ϕ ϕ, you could change it by rotation hence, you would not have spherical symmetry. If the charge density is only a function of r, that is ρ = ρ ( r ) ρ = ρ ( r ), then you have spherical symmetry. One good way to determine whether or not your problem has spherical symmetry is to look at the charge density function in spherical coordinates, ρ ( r, θ, ϕ ) ρ ( r, θ, ϕ ). In (c), the charges are in spherical shells of different charge densities, which means that charge density is only a function of the radial distance from the center therefore, the system has spherical symmetry. ![]() In (b), the upper half of the sphere has a different charge density from the lower half therefore, (b) does not have spherical symmetry. In (a), charges are distributed uniformly in a sphere. The spherical symmetry occurs only when the charge density does not depend on the direction. Charges on spherically shaped objects do not necessarily mean the charges are distributed with spherical symmetry. Different shadings indicate different charge densities. Therefore, this charge distribution does have spherical symmetry.įigure 6.21 Illustrations of spherically symmetrical and nonsymmetrical systems. Although this is a situation where charge density in the full sphere is not uniform, the charge density function depends only on the distance from the center and not on the direction. Thus, it is not the shape of the object but rather the shape of the charge distribution that determines whether or not a system has spherical symmetry.įigure 6.21(c) shows a sphere with four different shells, each with its own uniform charge density. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has uniform charge density ρ 1 ρ 1 and the bottom half has a uniform charge density ρ 2 ≠ ρ 1, ρ 2 ≠ ρ 1, then the sphere does not have spherical symmetry because the charge density depends on the direction ( Figure 6.21(b)). For instance, if a sphere of radius R is uniformly charged with charge density ρ 0 ρ 0 then the distribution has spherical symmetry ( Figure 6.21(a)). In other words, if you rotate the system, it doesn’t look different. Charge Distribution with Spherical SymmetryĪ charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. To exploit the symmetry, we perform the calculations in appropriate coordinate systems and use the right kind of Gaussian surface for that symmetry, applying the remaining four steps.
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