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The charge density inside a sphere of radius \( R \) is given by \[ \rho(r, \theta, \phi) = K r \cos(2\theta) \sin(\phi) \quad (K \text{ constant}) \] a. Calculate the dipole moment \(\vec{p}\) of the sphere. You may need the following integral: \[ \int \cos(2x) \sin^2(x) \, dx = \frac{1}{8}(-4x + 4\sin(2x) - \sin(4x)) \] b. Find the leading term in the multipole expansion of the potential at points far away from the sphere.

The charge density inside a sphere of radius \( R \) is given by \[ \rho(r, \theta, \phi) = K r \cos(2\theta) \sin(\phi) \quad (K \text{ constant}) \] a. Calculate the dipole moment \(\vec{p}\) of the sphere. You may need the following integral: \[ \int \cos(2x) \sin^2(x) \, dx = \frac{1}{8}(-4x + 4\sin(2x) - \sin(4x)) \] b. Find the leading term in the multipole expansion of the potential at points far away from the sphere.

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The charge density inside a sphere of radius \( R \) is given by \[ \rho(r, \theta, \phi) = K r \cos(2\theta) \sin(\phi) \quad (K \text{ constant}) \] a. Calculate the dipole moment \(\vec{p}\) of the sphere. You may need the following integral: \[ \int \cos(2x) \sin^2(x) \, dx = \frac{1}{8}(-4x + 4\sin(2x) - \sin(4x)) \] b. Find the leading term in the multipole expansion of the potential at points far away from the sphere.
Transcribed Image Text:The charge density inside a sphere of radius \( R \) is given by \[ \rho(r, \theta, \phi) = K r \cos(2\theta) \sin(\phi) \quad (K \text{ constant}) \] a. Calculate the dipole moment \(\vec{p}\) of the sphere. You may need the following integral: \[ \int \cos(2x) \sin^2(x) \, dx = \frac{1}{8}(-4x + 4\sin(2x) - \sin(4x)) \] b. Find the leading term in the multipole expansion of the potential at points far away from the sphere.
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