we have used spherical polar coordinates here
For the section of a sphere described by 1 ≤ ? ≤ 3, 90° ≤ ? ≤ 180°, 0° ≤ ? ≤ 60° , by expressing the respective integrations, find, (i) The total surface area of the spherical section (ii) The volume of the spherical section
In 2D, a sphere can be described by 2+sr. In 3D, a sphere can be described by r2 + y2 + .2-r*. We can talk about a sphere in n-dimensions by defining it like++2 Using what we know about multivariable calculus, believe it or not, it is relatively easy to calculate the volume of an n -dimensional sphere. It turns out that the volume of a 5th dimensional sphere of radius 1 will be a maximum, and then the volume...
(3) Let a > 0. In spherical coordinates, a surface is defined by r = 2a cos φ for 0 Find the volume of the solid enclosed by the surface, as a function of a. φ S (3) Let a > 0. In spherical coordinates, a surface is defined by r = 2a cos φ for 0 Find the volume of the solid enclosed by the surface, as a function of a. φ S
he surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r,0,o) by where θο is a constant (which you may assume is less than π/2 ) (a) Sketch the surface Sc. (b) Using the expression show that the vector element of area on Sc is given by where he surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r,0,o) by where θο is a constant (which you may assume is less than...
We were unable to transcribe this imageLet us denote the volume and the surface area of an n-dimensional sphere of adius R as V(OR)-VR and S(R)-S.),respectively (a) Find the relation between V(0) and S 1) (b) Calculate the Gaussian integral 3. (c) Calculate the same integral in spherical coordinates in terms of the gamma function re)-e'd (d) Obtain the closed forms of S,,(1) and V(1) (e) Calculate r5) and S.,0), p.(1) for n-1, 2, 3. (40 points) Let us denote...
2. (30 POINTS) A spherical shell of radius R holds a potential on its surface of: V(R, 0) = V.(1 + 2cose - cos20) (a.) Find the potential inside and outside the sphere. (b.) Find the surface charge density on the sphere. (c.) Find the dipole moment and the dipole term of the electric field, Epip.
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
please help with Q1 and 3 1. Let V be the solid region in R3 that lies within the sphere 2+y+z2-4, above the zy-plane, and below the cone z -Vx2 + y2 (a) Sketch the region V (b) Calculate the volume of V by using spherical coordinates. (c) Find the surface area of the part of V that lies on the sphere z2 y 24, by calculatinga surface integral. (d) Verify your solution to (c) by calculating the surface integral...
Use a spherical coordinate integral to find the volume of the given solid. sphere 0-1 and the cardioid of revolution o 5+ 2 cos p 21) the solid between the sphere o1 and the card Use a spherical coordinate integral to find the volume of the given solid. sphere 0-1 and the cardioid of revolution o 5+ 2 cos p 21) the solid between the sphere o1 and the card
Problem 2: a conducting sphere A conducting sphere has a positive net charge Q and radius R. (Note: since the sphere is conducting all the charge is distributed on its surface.) a) By reflecting on the symmetry of the charge distribution of the system, determine what the E-field lines look like outside the sphere for any r > R. Describe the E-field in words and with a simple sketch. Make sure to also show the direction of the E-field lines....