# Surface area of a shifted sphere in spherical coordinates

• MilkyWay2020
In summary, this problem can be solved using the spherical coordinate system and the sphere equation.
MilkyWay2020

## Homework Statement

find the surface area of a sphere shifted R in the z direction using spherical coordinate system.

## Homework Equations

$$S= \int\int \rho^2 sin(\theta) d\theta d\phi$$

$$x^2+y^2+(z-R)^2=R^2$$

## The Attempt at a Solution

I tried to use the sphere equation mentioned above and solve for ρ, using the definition of ρ and z in spherical coordinate system, this gives me:

$$x^2+y^2+z^2+R^2-2 R z= R^2$$
$$\rho^2-2 R \rho cos(\theta)= 0$$
$$\rho= 2 R cos(\theta)$$

then substitute in the integral:

$$S= \int\int 4 R^2 cos(\theta)^2 sin(\theta) d\theta d\phi$$

and integrate with the following boundaries:

θ: 0 → π/2
Φ: 0 → 2π

this gives a wrong answer of:

$$\frac{8 \pi R^2}{3}$$

What am I missing? This is a very basic problem but I cannot put my hand on the mistake.

MilkyWay2020 said:

## Homework Statement

find the surface area of a sphere shifted R in the z direction using spherical coordinate system.

## Homework Equations

$$S= \int\int \rho^2 sin(\theta) d\theta d\phi$$

$$x^2+y^2+(z-R)^2=R^2$$

## The Attempt at a Solution

I tried to use the sphere equation mentioned above and solve for ρ, using the definition of ρ and z in spherical coordinate system, this gives me:

$$x^2+y^2+z^2+R^2-2 R z= R^2$$
$$\rho^2-2 R \rho cos(\theta)= 0$$
$$\rho= 2 R cos(\theta)$$

then substitute in the integral:

$$S= \int\int 4 R^2 cos(\theta)^2 sin(\theta) d\theta d\phi$$
With your choice of variables shouldn't that be a ##\sin\phi## in the integral?

You need to recalculate the metric for the new parametrization. Your form of the integral implicitly assumes ##\rho=\text{constant}##.

Haborix said:
You need to recalculate the metric for the new parametrization. Your form of the integral implicitly assumes ##\rho=\text{constant}##.

I was afraid someone would post this specific answer. I had my doubts, but I thought the problem is too easy to get complicated. I think I was wrong about that.

Thank you, I am able to find the solution I am looking for.

berkeman

## 1. What is the formula for finding the surface area of a shifted sphere in spherical coordinates?

The formula for finding the surface area of a shifted sphere in spherical coordinates is A = 4πR2, where R is the radius of the sphere.

## 2. How do you convert Cartesian coordinates to spherical coordinates?

To convert Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ), you can use the following equations:

r = √(x2 + y2 + z2)

θ = arctan(y/x)

φ = arccos(z/r)

## 3. Can the surface area of a shifted sphere in spherical coordinates be negative?

No, the surface area of a shifted sphere in spherical coordinates cannot be negative. It is always a positive value.

## 4. How does the surface area of a shifted sphere change as the radius increases?

The surface area of a shifted sphere increases as the radius increases. This is because the surface area is directly proportional to the square of the radius.

## 5. Is there a difference between the surface area of a shifted sphere in spherical coordinates and a regular sphere?

Yes, there is a difference between the surface area of a shifted sphere in spherical coordinates and a regular sphere. A shifted sphere has its center shifted from the origin, while a regular sphere has its center at the origin. This results in a difference in the coordinates and therefore, a difference in the surface area calculation.

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