Find the volume of the torus centered at the origin whose tube radius is 1 and...

Find the volume of the torus centered at the origin whose tube radius is 1 and whose distance from the origin to the center circle is 4. （By Change variables)

Expert Solution

Colby Messinger answered 2 years ago

Let S ∈ R3 be the sphere of radius 1 centered on the origin. a) Prove...

Let S ∈ R3 be the sphere of radius 1 centered on the origin. a) Prove that there is at least one point of S at which the value of the x + y + z is the largest possible. b) Determine the point (s) whose existence was proved in the previous point, as well as the corresponding value of x + y + z.

A solid is bounded by the sphere centered at the origin of radius 5 and the...

A solid is bounded by the sphere centered at the origin of radius 5 and the infinite cylinder along the z-axis of radius 3. (a) Write inequalities that describe the solid in Cartesian coordinates. (b) Write inequalities that describe the solid in cylindrical coordinates. (c) Why is this solid difficult to describe in spherical coordinates? Which of the variables ρ, θ, φ are difficult to describe? Explain.

A hemisphere of radius R is centered on the origin and immersed in an electric field,...

A hemisphere of radius R is centered on the origin and immersed in an electric field, E, given by E = (B cos(θ) / r) r + Ar^2 sin^2 (θ) θ + Cr^3 cos^2 (θ) φ. Find the charge enclosed in the hemisphere

Acircular ring of radius ?lies in the ??plane and is centered on the origin. The half...

Acircular ring of radius ?lies in the ??plane and is centered on the origin. The half on the positive ?side is uniformly charged with a charge +?while the half on the negative ?side is uniformly charged with a total charge −?. a..Draw a diagram of the charge distribution and without doing any math, determinethe direction of the total electric field at the origin . b.Calculate the ?component of the electric field at the origin by integrating the charged ring. i.Draw...

A ring of charge with radius R = 2.5 m is centered on the origin in...

A ring of charge with radius R = 2.5 m is centered on the origin in the x-y plane. A positive point charge is located at the following coordinates: x = 17.1 m y = 3.8 m z = -16.3 m The point charge and the total charge on the ring are the same, Q = +81 C. Find the net electric field along the z-axis at z = 4.5 m. Enet,x = Enet,y = Enet,z =

a) Let D be the disk of radius 4 in the xy-plane centered at the origin....

a) Let D be the disk of radius 4 in the xy-plane centered at the origin. Find the biggest and the smallest values of the function f(x, y) = x 2 + y 2 + 2x − 4y on D. b) Let R be the triangle in the xy-plane with vertices at (0, 0),(10, 0) and (0, 20) (R includes the sides as well as the inside of the triangle). Find the biggest and the smallest values of the function...

A solid sphere of charge is centered at the origin and has radius R = 10...

A solid sphere of charge is centered at the origin and has radius R = 10 cm. Instead of being uniformly charged, the charge density varies with radial position: ρ(r)=ρ0ar. Take a=5.1 m and ρ0=3.7 C/m3. What is the total charge of the sphere? What is the electric flux through a sherical surface of radius R/2 that is concentric with the charged sphere? What is the flux through a spherical surface of radius 2R that surrounds the charged sphere, but...

A solid sphere, radius R, is centered at the origin. The “northern” hemisphere carries a uniform...

A solid sphere, radius R, is centered at the origin. The “northern” hemisphere carries a uniform charge density ρ0, and the “southern” hemisphere a uniform charge density −ρ0. Find the approximate field E(r,θ) for points far from the sphere (r ≫ R). P.S. Please be as neat as possible.

3: Find the volume of the torus shown using: a. The Shell Method. b. The Washer...

3: Find the volume of the torus shown using: a. The Shell Method. b. The Washer Method.

Consider a long, thin, plastic cylindrical shell centered at the origin. It has a radius 2R...

Consider a long, thin, plastic cylindrical shell centered at the origin. It has a radius 2R and a linear charge density -3 λ . b. Use Gauss's Law to find the electric field (mag and direction) at a distance x=3R from the origin. Now a long line of charge (parallel to the axis of the cylinder) is added at a distance x=4R with linear charge density + λ . c. use superposition to find the force (mag. and direction) on...

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