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(16 pts total) The potential at the surface of a sphere of radius R is given...
5. A hollow sphere of radius R has a potential on the surface of V(θ, d)...
5. A hollow sphere of radius R has a potential on the surface of V(θ, d) Vo cos θ. There is no a) Find the potential everywhere inside and outside the sphere. b) Find the electric field everywhere inside the sphere. (You will find it easier to convert the potential to Cartesian coordinates and then find the field.) c) Find the charge density σ(0) on the surface of the sphere using Gauss' law. charge inside or outside the sphere.
1. The potential at the surface of a sphere is kept at potential V(R.0)-Vo sin20. The...
1. The potential at the surface of a sphere is kept at potential V(R.0)-Vo sin20. The potential at infinity is zero. (a) Find V(r, 0) inside the sphere. (b) Find V(r,0) outside the sphere. (c) Find σ(θ), the charge density on the sphere. (d) Find the total charge of the sphere. (e) The problern would be a lot harder if the potential were specified to be V(R,θ)-Võsin θ Why? Explain how you would do part (a) without going through the...
The potential at the surface of a sphere is given by V = kcos(theta) where k...
The potential at the surface of a sphere is given by V = kcos(theta) where k is a constant. A) find the potential inside and outside the sphere (no charge present inside or outside the sphere) B) Determine the charge density sigma on the surface of the sphere.
6. The electric potential at the surface of a sphere of radius R is constant, i.e.,...
6. The electric potential at the surface of a sphere of radius R is constant, i.e., V(R,0) = k, where k + 0. Very far away from the sphere (r >> R) the electric potential is V(r,0) = kr cos(0). Find the electric potential outside the sphere, remember to check that your answer matches the boundary conditions (1 point).
Determine the electric potential outside (r > R) of a metal sphere of radius R divided up into...
Determine the electric potential outside (r > R) of a metal sphere of radius R divided up into hemispheres, where the upper hemisphere ( 0 ≤ θ ≤ π/2 ) is held at potential V, and the lower hemisphere (π/2 < θ ≤ π ) is grounded (held at zero potential). This is identical to a problem worked out in class, except for the region of interest. Express all coefficients in terms of Legendre polynomials. Do not leave any in...
A conducting sphere of radius a, at potential Vo, is surrounded by a thin concentric spherical...
A conducting sphere of radius a, at potential Vo, is surrounded by a thin concentric spherical shell of radius b, over which someone has glued a surface charge density σ(8)-k cos θ, where k is a constant and θ is the polar spherical coordinate. (a) Find the potential in each region: (i)r > b, and () a<r<b. [5 points] [Hint: start from the general solution of Laplace's equation in spherical coordinates, but allow for different coefficients in the radial part...
The potential at the surface of a sphere is given by Vo(0)-kcos 30 Find the potential...
The potential at the surface of a sphere is given by Vo(0)-kcos 30 Find the potential inside and outside the sphere, as well as the surface charge density ?(0) on the sphere. (Assume there is no other charge in the problem except on the sphere.) Make sure to evaluate the coefficients using Fourier's "trick", rather than just guessing. Make a sketch of the electric field.
2. Potential Inside a Sphere We are interested in the electric potential inside a spherical shell...
2. Potential Inside a Sphere We are interested in the electric potential inside a spherical shell that is radius a and centered on the origin. There are no charges inside the she, so the potential satisfies the Laplace equation, However, there is an external voltage applied to the surface of the shell which holds the potential on the surface to a value which depends on θ: As a result, the potential Ф(r,0) -by symmetry, it does not depend on ф-is...
5. Charge distributed on a spherical surface of radius a produces the potential φ(a, 0) φ.cos)...
5. Charge distributed on a spherical surface of radius a produces the potential φ(a, 0) φ.cos) on that surface, with θ the polar angle and φ, constant. Expressing answers in terms of the givens only, (a) Find φ(r,0), inside the surface and outside (both charge-free). Use zonal harmonics: Eq (3.65), pg 143. (b) Find the surface-charge density function σ(0). (Recall o-e,AE,ORMAL.) (c) Evaluate the dipole moment of the charge distribution, by comparing your exterior solution in (a) to the standard...
Charge is spread uniformly over the surface of a sphere of radius R. The potential at...
Charge is spread uniformly over the surface of a sphere of radius R. The potential at the sphere's center is V. Find an expression for the net charge Q on the sphere. Express your answer in terms of the variables R, V, and the Coulomb's constant k.
5. A hollow sphere of radius R has a potential on the surface of V(θ, d) Vo cos θ. There is no a) Find the potential everywhere inside and outside the sphere. b) Find the electric field everywhere inside the sphere. (You will find it easier to convert the potential to Cartesian coordinates and then find the field.) c) Find the charge density σ(0) on the surface of the sphere using Gauss' law. charge inside or outside the sphere.
1. The potential at the surface of a sphere is kept at potential V(R.0)-Vo sin20. The potential at infinity is zero. (a) Find V(r, 0) inside the sphere. (b) Find V(r,0) outside the sphere. (c) Find σ(θ), the charge density on the sphere. (d) Find the total charge of the sphere. (e) The problern would be a lot harder if the potential were specified to be V(R,θ)-Võsin θ Why? Explain how you would do part (a) without going through the...
6. The electric potential at the surface of a sphere of radius R is constant, i.e., V(R,0) = k, where k + 0. Very far away from the sphere (r >> R) the electric potential is V(r,0) = kr cos(0). Find the electric potential outside the sphere, remember to check that your answer matches the boundary conditions (1 point).
A conducting sphere of radius a, at potential Vo, is surrounded by a thin concentric spherical shell of radius b, over which someone has glued a surface charge density σ(8)-k cos θ, where k is a constant and θ is the polar spherical coordinate. (a) Find the potential in each region: (i)r > b, and () a<r<b. [5 points] [Hint: start from the general solution of Laplace's equation in spherical coordinates, but allow for different coefficients in the radial part...
The potential at the surface of a sphere is given by Vo(0)-kcos 30 Find the potential inside and outside the sphere, as well as the surface charge density ?(0) on the sphere. (Assume there is no other charge in the problem except on the sphere.) Make sure to evaluate the coefficients using Fourier's "trick", rather than just guessing. Make a sketch of the electric field.
2. Potential Inside a Sphere We are interested in the electric potential inside a spherical shell that is radius a and centered on the origin. There are no charges inside the she, so the potential satisfies the Laplace equation, However, there is an external voltage applied to the surface of the shell which holds the potential on the surface to a value which depends on θ: As a result, the potential Ф(r,0) -by symmetry, it does not depend on ф-is...
5. Charge distributed on a spherical surface of radius a produces the potential φ(a, 0) φ.cos) on that surface, with θ the polar angle and φ, constant. Expressing answers in terms of the givens only, (a) Find φ(r,0), inside the surface and outside (both charge-free). Use zonal harmonics: Eq (3.65), pg 143. (b) Find the surface-charge density function σ(0). (Recall o-e,AE,ORMAL.) (c) Evaluate the dipole moment of the charge distribution, by comparing your exterior solution in (a) to the standard...
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