The potential at the surface of a sphere (radius R) is given by
V_0 = k cos 3theta
where k is constant. Find the potential inside and outside the sphere as well as the surface charge density (lower case sigma(theta)) on the sphere. (assume there is no charge inside or outside the sphere.
To find the potential inside and outside the sphere, we need to apply the appropriate boundary conditions at the surface of the sphere.
Inside the sphere (r < R):
The potential can be determined by using Laplace's equation, which states that the Laplacian of the potential is zero (ΔV = 0). In spherical coordinates, the Laplacian is given by:
ΔV = (1/r^2) ∂(r^2 ∂V/∂r) + (1/r^2 sin θ) ∂/∂θ (sin θ ∂V/∂θ) + (1/r^2 sin^2 θ) ∂^2V/∂φ^2
Since there is no charge inside the sphere, we know that the potential is constant and equal to V_0 at the surface of the sphere (r = R). Therefore, we can write:
V(R) = V_0 = k cos 3θ
To solve the Laplace's equation inside the sphere, we can assume a solution in terms of Legendre polynomials. By applying appropriate boundary conditions, it can be shown that the general solution for the potential inside the sphere is given by:
V(r, θ) = ∑[A_l r^l + B_l r^-(l+1)] P_l(cos θ)
In this case, since there is no charge inside the sphere, we can disregard the terms containing B_l.
Outside the sphere (r > R):
Similarly, the potential outside the sphere is also a solution to Laplace's equation, but subject to different boundary conditions. Since there is no charge outside the sphere, we know that the potential should approach zero as r goes to infinity. Therefore, the general solution for the potential outside the sphere is given by:
V(r, θ) = ∑[C_l r^-(l+1)] P_l(cos θ)
In this case, we don't need to consider the terms containing A_l because they would violate the boundary condition at r = R.
Surface Charge Density:
The surface charge density (σ) on the sphere can be determined using the relationship between the electric field (E) and the potential (V). The electric field is given by the negative gradient of the potential:
E = -∇V
Since the potential depends only on θ in this case, the electric field can be written as:
E = -∂V/∂r r̂ - (1/r) ∂V/∂θ θ̂ - (1/r sin θ) ∂V/∂φ φ̂
= (1/r) ∂V/∂θ θ̂
At the surface of the sphere (r = R), the electric field is related to the surface charge density by Gauss's law:
E = σ/ε0
Substituting the expression for the electric field, we have:
(1/R) ∂V/∂θ = σ/ε0
Using the given potential, we can find the surface charge density:
σ = (ε0/R) ∂V/∂θ
= (3ε0k/R) sin 3θ
To find the potential inside and outside the sphere, we need to relate the potential to the charge density on the surface of the sphere.
Let's start with finding the potential inside the sphere. Inside the sphere, the potential is only determined by the charge distribution on the surface. We can write the potential, V, inside the sphere as:
V = ∫[0 to R] k' cos3θ' σ(θ') R^2/(r'^2) dθ'
where θ' is the angle made by a point on the surface relative to a reference point inside the sphere, k' is a new constant that incorporates the original constant k, σ(θ') is the surface charge density as a function of θ', r' is the distance from the charge element on the surface to the point inside the sphere, and R is the radius of the sphere.
Now, since there is no charge inside the sphere, the electric field inside the sphere is zero. Therefore, the electric field is the gradient of the potential:
E = -∇V
Inside the sphere, the potential is only dependent on θ', so we can express the gradient as:
∇V = ∂V/∂θ' * (r'^-1) * ∂/∂θ'
Since E = -∇V and E = 0 inside the sphere, we have:
∂V/∂θ' = 0
This means that the potential inside the sphere does not vary with θ'. Therefore, the potential inside the sphere is constant and equal to V_0.
Now let's find the potential outside the sphere. Outside the sphere, the potential is determined both by the charge distribution on the surface and the charge distribution in the surrounding space. Similar to the approach for finding the potential inside, we can write the potential, V, outside the sphere as:
V = ∫[0 to R] k' cos3θ' σ(θ') r'^2 dθ' / r'
where r' is the distance from the charge element on the surface to the point outside the sphere.
Since there is no charge outside the sphere, the electric field, E, is the gradient of the potential:
E = -∇V
Using the same reasoning as before, we can find that the potential outside the sphere is given by:
V = - k' ∫[0 to R] cos3θ' σ(θ') r'^2 dθ' / r'
To find the surface charge density (σ(θ')), we can use the definition of electric field at the surface of the sphere. At the surface of the sphere, the electric field must be continuous, and since E = -∇V at the surface, we have:
E_surface = -∇V = - (∂V/∂r')(r=R)
The electric field at the surface can also be expressed as:
E_surface = -σ / ε₀
where σ is the surface charge density and ε₀ is the permittivity of free space.
Combining the two expressions for the electric field at the surface, we have:
- σ / ε₀ = - (∂V/∂r')(r=R)
Differentiating the potential V with respect to r', we get:
(∂V/∂r') = -k' ∫[0 to R] 3cos3θ' σ(θ') r' dθ' / r'^2
Evaluating (∂V/∂r')(r=R) and setting it equal to - σ / ε₀, we get:
- σ / ε₀ = -k' ∫[0 to R] 3cos3θ' σ(θ') dθ' / R^2
Simplifying, we find:
σ(θ') = (3ε₀ k' / R^2) cos3θ'
Therefore, the surface charge density on the sphere is proportional to cos3θ'.
To summarize:
- The potential inside the sphere is constant and equal to V_0.
- The potential outside the sphere is given by V = - k' ∫[0 to R] cos3θ' σ(θ') r'^2 dθ' / r'.
- The surface charge density on the sphere is given by σ(θ') = (3ε₀ k' / R^2) cos3θ'.
http://www.iitg.ernet.in/physics/fac/charu/courses/ph410_2007/tut4/node1.html
See prob 21