What is Spherical charge distribution: Definition and 17 Discussions

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
Spherical harmonics originates from solving Laplace's equation in the spherical domains. Functions that solve Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree

{\displaystyle \ell }


{\displaystyle (x,y,z)}
that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence


{\displaystyle r^{\ell }}
from the above-mentioned polynomial of degree

{\displaystyle \ell }
; the remaining factor can be regarded as a function of the spherical angular coordinates


{\displaystyle \theta }


{\displaystyle \varphi }
only, or equivalently of the orientational unit vector


{\displaystyle {\mathbf {r} }}
specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition.
A specific set of spherical harmonics, denoted




{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )}






{\displaystyle Y_{\ell }^{m}({\mathbf {r} })}
, are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.
Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes.

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  1. A

    B Can charges move without a field? How charges redistribute without it?

    So from Gauss theorem, electric field at any point inside a uniformly charged sphereical shell is zero. Thus there is no electrostatic force on the inner sphere. From what I have learnt, a field is necessary to move charges. But in this case the inner sphere acquires a charge q without any...
  2. Mohomad

    Spherical charge distribution to generate this E-field

    A spherical volume charge (R<=1cm) with uniform density ρv0 is surrounded by a spherical surface charge ( R=2cm) with charge density 4 C/m2. If the electric field intensity at R=4cm is 5/Є0 ,deterime ρv0
  3. jlmccart03

    Magnitude of electric field near spherical shell

    Homework Statement A 10-nC point charge is located at the center of a thin spherical shell of radius 8.0 cm carrying -20 nC distributed uniformly over its surface. What is the magnitude of the electric field 2.0 cm from the point charge? Homework Equations E = kq1q2/r^2 The Attempt at a...
  4. J

    Electric field from spherical charge distribution

    Hello. I have a problem calculating the electric field from spherical charge distribution. The exercise is: 1. Homework Statement Homework Equations To solve the problem for $$ 0\le R < a$$ i tried 2 ways: $$ \vec{E} = \frac{\vec{a_R}}{4\pi\epsilon_0}\int_v\frac{1} {R^2}\rho dv $$ and the...
  5. P

    Using Gauss' law for spherical charge distribution

    Homework Statement The Earth is constantly being bombarded by cosmic rays, which consist mostly of protons. Assume that these protons are incident on the Earth’s atmosphere from all directions at a rate of 1366. protons per square meter per second. Assuming that the depth of Earth’s atmosphere...
  6. P

    Charge distribution in the universe

    Homework Statement Can we consider the universe to have a uniformly charged distribution? If so, shouldn't the field at any point in space be zero? Since the universe is infinite, will it be symmetrical about any point, field should be zero right? Why is this not true?[/B]2. The attempt at a...
  7. H

    Spherical Charge distribution

    Homework Statement A Non-Uniform but spherically symmetric charge distribution has a charge density: \rho(r)=\rho_0(1-\frac{r}{R}) for r\le R \rho(r)=0 for r > R where \rho = \frac{3Q}{\pi R^3} is a positive constant Show that the total charge contained in this charge distribution is...
  8. deedsy

    Spherical Charge Distribution - Electric Field Intensity

    Homework Statement A spherical charge distribution is given by p = p_0 (1- \frac{r^2}{a^2}), r\leq a and p = 0, r \gt a , where a is the radius of the sphere. Find the electric field intensity inside the charge distribution. Well I thought I found the answer until I looked at the back of...
  9. O

    Potential of spherical charge distribution

    I want to derive this equation: V(r) = \frac{1}{\epsilon_0} [\frac{1}{r} \int_0^r \! r'^2 \rho(r') \, d r' + \int_r^{\infty} \! r' \rho(r') \, d r' ] of a spherical charge distribution. I can do it with the general integral definition of the electrostatic potential (which is basically...
  10. Avatrin

    Coulomb's law and spherical charge distribution

    Homework Statement Find the E produced by a spherical charge distribution with uniform charge density at a point inside the sphere, using triple integration. Homework Equations E = 1/4πε ∫f(x,y,z)/r^2 dV The Attempt at a Solution f(x,y,z) = p Radius of sphere = R Position of...
  11. P

    Electric Potential of a Spherical Charge Distribution

    Homework Statement A total charge q is uniformly distributed throughout a sphere of radius a. Find the electric potential in the region where r1<a and r2>a. The potential is defined anywhere inside the sphere. Homework Equations letting ρ = volume charge density and ε = permittivity...
  12. K

    How do I Calculate Electric Field of a Spherical Charge Distribution?

    good evening! i am trying to calculate the electric field of a spherical charge distribution ρ=ρ_{0}e^{-kr}, where r is the radial distance. i am a little bit embarressed,but i have to say that i am not comfortable with spherical coordinates in practical calculations. i would appreciate if...
  13. T

    Calculating Electric Field of Spherical Charge Distribution

    Homework Statement Compute the electric field generated by a spherically symmetric charged sphere of radius R with charge density of \rho = kr^{2} Homework Equations \oint _S \vec{E} \cdot \vec{dA} = \frac{Q_{enclosed}}{\epsilon_0} The Attempt at a Solution I know that this question...
  14. A

    Gauss law and infinite spherical charge distribution

    1. The problem statement Consider an infinite spherical charge distribution with constant charge density. According to symmetry of the problem, I expect the electric field at any point to be zero. But if you construct a Gaussian sphere and apply Gauss theorem, it will give you some finite field...
  15. F

    Gauss' Law, Spherical Charge Distribution

    Before I get into the question I'd just like to state that this is not homework, but questions in my book that I'm going through to prepare myself for the midterm in one week. I got stuck at a few questions, here's the first one. I won't ask the next until I'm done with this and so forth...
  16. J

    Finding Electric Field for Spherical Charge Distribution?

    Homework Statement There is a charge density rho that exists in a spherical region of space defined by 0 < r < a. \rho (r) = Ke^{-br} How do you find the electric field if a charge density varies as such?The Attempt at a Solution I found Q total = \int \int \int \rho dV Now I need to find E...
  17. J

    Determining Electric field of Spherical charge distribution

    Hey, this is from "Foundations of Electromagnetic Theory" by Reitz, et al. Problem 2-15. I have had a really hard time trying to learn from this book as there are no examples to apply the equations they prove throughout the chapters. Anyhow, I don't really have anything down for this problem...