He surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r,0,o) by ...
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...
In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical symmetry about the polar axis is bounded on the polar axis can be expressed as u(r, 0) = Rm(r)P,(cos 0), (A) where P is the Legendre polyomial of degree n, and R(r) is the general solution of the differential equation *() - n(n + 1)R = 0, (r > 0), dr dr where n is a non-negative integer. (You are not asked to show...
Write the vector differential operator "DEL-V in Cartesian coordinates Cylindrical coordinates Spherical coordinates. 2. Show for any "nice" scalar function (x,y,z), the Curl of the gradient of (x,y,z) is Zero.. VxVo = 0 Hint: assume the order of differentiation can be switched 3. Find the volume of a sphere of radius R by integrating the infinitesimal volume element of the sphere. 4. Write Maxwell's equations for the case of electro and magneto statics (the fields do not change in time)...
2. Potentials and a Conducting Surface The electric potential outside of a solid spherical conductor of radius R is found to be V(r, 9) = -E, cose (--) where E, is a constant and r and 0 are the spherical radial and polar angle coordinates, respectively. This electric potential is due to the charges on the conductor and charges outside of the conductor 1. Find an expression for the electric field inside the spherical conductor. 2. Find an expression for...
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
Suppose that a scalar field is constant on a surface As shown in the lectures. there are two methods that one might use to obtain the normal to the surface, and they give the same direction (a) Let r(u, v) be a parametric form for the surface S. Use the vector identity to show that Our ar-λ▽u where λ is a scalar field. [Note: no marks will be awarded for simply stating that a term is zero. If it is...
2014/B5 (a) Draw skecthes to illustrate R, 0 and z coordinate curves for the case of cylindrical polar coordinates (b) Show that the gradient of a scalar field, p, can be expressed in terms of curvilinear coordinates u1, u2 and us, of an orthogonal coordinate system as where h, Idr/dul. Hence obtain a formula for Vip in cylindrical polar coordinates. (c) Evaluate dp/ds, the rate of change of φ with distance, for the field φ-R, cost) at the point R...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addition, show the unit vectors r, θ, and φ at that point B. Write the vectors ŕ.0, and ф in terms of the unit vectors x, y, and г. Here's the easy way to do this 1. For r, simply use the fact that/r 2. For φ, use the following formula sin θ Explain why the above formula works 3. Compute θ via θ...
A spherical balloon has a radius o R-70 cm, and its surface has been charged uniform a) Calculate the field strength ri 54 cm from the balloon's centre b) Calculate the field strength r2-191 cm from the balloon's centre (c)Calculate the net charge on the balloon. You measure the elec c field to be = at he balloons sur ace where N PAPER SOLUTION Solve the problem on paper first, including all four IDEA steps. You will become a better...
. (40 points: A membrane is stretched under tension r with uniform surface density o. (Small-amplitude) transverse displacements of this stretched membrane satisfy the wave equation కా - ఆ (10) Suppose the membrane is stretched in a rectangular frame lying in the plane z= 0, with side-lengths a, b. mitially stationary at all points, the membrane is struck at t-0 and thus set vibrating- 3.1. Show that e satisfies the following expression: veu)-ΣΣ..-ΣΣ Β.in may sin sin (wt] %3D (11)...