Question

In: Physics

Consider a conducting hollow sphere with radius R that is placed in a homogeneous electric field...

Consider a conducting hollow sphere with radius R that is placed in a homogeneous electric field E_0 = E_0 e_z

a) Calculate the electrostatic potential φ_0(r) for the homogeneous electric field E_0= E_0 e_z only and write the result in spherical coordinates.

b) Assume that the sphere is grounded i.e. put the potential φ(R)=0 and calculate the electrostatic potential φ(r)=0 inside and outside the sphere.

Hint: Consider that the electrostatic potential far away from the sphere should just give rise to the homogeneous electric field E_0, i.e. potential should have this limit at large distances.

c) Determine the surface charge density ρ(θ,φ) on the sphere.

Hint: The solution has the form ρ(θ,φ)& ρ’(cos(θ)).

d) Calculate the dipole moment p=∫r φ(r) dV of the induced charge density on the surface of the sphere where φ(r) ist the charge density.

Expert Solution

Solving Laplace equation

a) and b)

image method

genius_generous answered 2 years ago

(a) Plot the electric field of a charged conducting solid sphere of radius R as a...

(a) Plot the electric field of a charged conducting solid sphere of radius R as a function of the radial distance r, 0 < r < 1, from the center. (b) Plot the electric field of a uniformly charged nonconducting solid sphere of radius R as a function of the radial distance r, 0 < r < 1, from the center.

Consider a hollow, infinite sphere of radius R. the hollow space is free of charge, but...

Consider a hollow, infinite sphere of radius R. the hollow space is free of charge, but a surface charge sigma = sigma not cos theta exists on the inside surface of the conductor at s =R. Can this sphere be a conductor? ( I.e, can you induce this charge on a conducting surface somehow)

A hollow, conducting sphere with an outer radius of .250 m and an inner radius of...

A hollow, conducting sphere with an outer radius of .250 m and an inner radius of .200 m has a uniform surface charge density of -6.37 muC/me2.When a charge is now introduced at the center of the cavity inside the sphere, the new charge density on the outside of the sphere is -4.46 muC/me2. What is the charge at the center of the cavity?

A hollow, conducting sphere with an outer radius of 0.240 m and an inner radius of...

A hollow, conducting sphere with an outer radius of 0.240 m and an inner radius of 0.200 m has a uniform surface charge density of +6.37 × 10−6 C/m2. A charge of -0.400 μC is now introduced into the cavity inside the sphere. What is the new charge density on the outside of the sphere? Express your answer with the appropriate units. Calculate the strength of the electric field just outside the sphere. Express your answer with the appropriate units....

A solid sphere of radius R, a solid cylinder of radius R, and a hollow cylinder...

A solid sphere of radius R, a solid cylinder of radius R, and a hollow cylinder of radius R all have the same mass, and all three are rotating with the same angular velocity. The sphere is rotating around an axis through its center, and each cylinder is rotating around its symmetry axis. Which one has the greatest rotational kinetic energy? both cylinders have the same rotational kinetic energy the solid cylinder the solid sphere they all have the same...

A hollow sphere of mass M and radius R is hit by a pendulum of mass...

A hollow sphere of mass M and radius R is hit by a pendulum of mass m and length L that is raised at an angle θ. After the collision the pendulum comes to a stop and the sphere rolls forward into a spring with stiffness k on an incline of ϕ. Find an expression for how far ∆x the spring compresses along the incline. Use only m, L, M, R, θ, ϕ, k and appropriate constants.

A non-conducting sphere of radius R centered at O contains a spherical cavity of radius R’...

A non-conducting sphere of radius R centered at O contains a spherical cavity of radius R’ centered at O'. Let d be the displacement of O’relative to 0. Throughout the sphere, there is a uniform charge density rho_0 (except inside the cavity, which is uncharged). (a) Use the principle of superposition to write down an expression for E(r) everywhere. (b) Repeat (a) for the electric potential b(r).

Consider a conducting sphere of radius R carrying a net charge Q. a). Using Gauss’s law...

Consider a conducting sphere of radius R carrying a net charge Q. a). Using Gauss’s law in integral form and the equation |E| = σ/ε0 for conductors, nd the surface charge density on the sphere. Does your answer match what you expect? b). What is the electrostatic self energy of this sphere? c). Assuming the sphere has a uniform density ρ, what is the gravitational self energy of the sphere? (That is, what amount of gravitational energy is required/released when...

A conducting sphere with a radius of R = 14.9 mm has a uniform and constant...

A conducting sphere with a radius of R = 14.9 mm has a uniform and constant surface charge density of a = 4.7 nC / m ?. What will be the magnitude of the electric field produced by that sphere at a distance from the center of the sphere of r = 34.3 cm?

Find the electric field inside and outside of hallow conducting sphere charge Q