A spherical shell of radius R centered about the origin carrying a uniform surface charge o spins at an angular velocity o = (ê sin y +î cos y) w. To evaluate the vector potential A(0,0, z) (coordinates in Cartesian coordinates) in Coulomb gauge, we %3D can evaluate 27 Ã(0,0,2) = / « dcos 0' o' dợ'R*Ÿ (e', o') (1) where one recognizes dcos 0'do'R² as an area element on the spherical shell. What is Ý before doing any of the integration? [Express your answer in terms of {@,e', ø', y,R, z,£, §, ¿}].
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- Suppose that we have a conductive sphere to the radius ( a ) containing the specific potential ( v0 ) this sphere we positioned in the center of a charged ring to the radius ( b ) , solve the problem of the total Q loop using the potential images of this system at a point such as P .In an electrostatic boundary value problem, the surface charge density o is specified at x = 0. The corresponding boundary condition for potential V can be written as 1 o(ya) 1 Lo(yz) (a) (b) x=0 x=0 Əv (c) ´av (d). 1 1 -6(yz) o(y.z) =+- x=0A thin-shelled hollow sphere of radius R has a uniform surface charge density σ. Fixed at its center is a point charge q. a) Use Gauss's law to find the electric field a distance r from the center. (Give answers for r < R and r > R.) b) Taking the electric potential V to vanish at infinity, find the electric potential as a function of r, the distance from the center. (Give answers for r < R and r > R.)
- If the electrostatic potential in spherical polar coordinates is where , and n are constants, then the charge density at a distance r=, will be EoPo (a) (b) 217 (c) erő (d) ero 2 er?Suppose we have a 2-dim. electrostatic field which varies with the coordinates x and y but is independent of z. Show that the average value of the potential on any circle parallel to the xy plane equals the potential at the center of the circle, provided the charge density in the region is zero.Only 2 c)
- The space between spherical conducting shells r = 3 cm and r = 7 cm is filled with a dielectric material for which E=2.5EoThe two shells are maintained at a potential difference of 75V. Find the capacitance of the system, and also caluclate charge density on shell r= 3 cm.A non-conducting sphere of radius R = 5 cm has its center at the origin O of coordinate system as shown in Fig. Its has two spherical cavities of radius r = 1 cm, whose centers are at (0, 3 cm), (0. -3 cm), respectively, and solid material of the sphere has uniform positive charge density p = 1/7 µСm-³. Calculate electric potential at point P (4 cm, 0). y Fig. P XConsider a grounded, conducting, spherical shell of outer radius b and inner radius a. Using the method of images, dicuss the problem of a point charge q inside the shell, i.e. at a distance r<a from the center. Find a) the potential inside the sphere; b) the induced surface charge density on the inner surface of the shell at r=a; what is the total induced charge? c) the magnitude and direction of the force acting on q. Does q get pushed towards the center, or away from the center? d) Is there any change in the solution if the sphere is kept at a fixed potential φo? If the sphere has a fixed total charge Q?
- A dipole with –Q at the origin (0,0) and the +Q at coordinates x= 4.00 cm, y = 3.00 cm is in an Electric field, E = 4.25 V/m directed to the right, +x . (Q = 0.400 C) Find the dipole moment.A dielectric sphere in an external field. Consider a simple dielec- tric with permittivity e, in the form of a uniform spherical ball of radius a. It is placed at the origin in an external electrostatic potential (x, y, z) = bxy (where r, y, z are Cartesian coordinates and b is a constant). Find the elec- trostatic potential o and electric field E everywhere. %3DConsider two spherical infinitely thin conducting surfaces (= ∞o) with the same center. The cross section of this configuration is shown below. The inner sphere (radius a) has a total negative charge of -Q. The outer sphere (radius b) has a positive charge with unknown amount. Assume the electric potential at the center is Vo (Vo > 0) and at infinity is zero. Also, assume free-space permittivity in all regions. (a) Find the electric potential V(R) at the distance R from the center for 0 < R < b. (b) Find the amount of positive charge on the outer sphere. Hint: The integral form of Gauss' Law is your best friend! Also you may pay close attention to the net charge.