How Do You Calculate the Resistance of a Hollow Sphere?

[lockedup]

Homework Statement

Determine a formula for the total resistance of a spherical shell made of material whose conductivity is $$\sigma$$ and whose inner and outer radii are r₁ and r₂. Assume the current flows radially outward.

Homework Equations

$$\sigma$$ = 1/$$\rho$$

R = ($$\rho$$*l)/A

The Attempt at a Solution

Since the current is moving outward, l (length) is going to be the difference of the radii. How do I account for the changing surface area?

The solution is (r₁ - r₂)/(4$$\pi$$*$$\sigma$$*r₁*r₂)

EDIT: *facepalm* I spelled hollow wrong...

 

[anathema]

To account for the changing surface area, we can use the formula for the surface area of a spherical shell, which is given by:

A = 4π(r2^2 - r1^2)

Substituting this into the formula for resistance (R = ρl/A), we get:

R = (ρl)/(4π(r2^2 - r1^2))

Since ρ = 1/σ, we can rewrite this as:

R = (1/σ)(l)/(4π(r2^2 - r1^2))

Substituting l = r2 - r1, we get the final formula for the total resistance of a spherical shell:

R = (r2 - r1)/(4πσr1r2)

This matches the solution given in the forum post.

 

[tomcue]

It should be resistance of a hollow sphere. Sorry about that!

I would approach this problem by first considering the basic principles of electrical resistance. Resistance is defined as the hindrance to the flow of electrical current through a material. It is dependent on the material's conductivity (σ) and the physical dimensions of the object through which the current is flowing.

In this case, we are dealing with a spherical shell, which means we can use the formula for resistance of a cylindrical shell, since both shapes have a similar surface area. The formula for resistance in a cylindrical shell is R = (ρ*l)/A, where ρ is the resistivity of the material, l is the length of the cylinder, and A is the cross-sectional area.

However, in this case, the length (l) is not constant as the current is flowing radially outward, so we need to account for this changing length. We can do this by taking the difference between the inner and outer radii of the spherical shell, which gives us the length of the current path. So our new formula becomes R = (ρ*(r2-r1))/A.

Now, we need to determine the cross-sectional area (A). Since we are dealing with a spherical shell, we can use the formula for the surface area of a sphere, which is 4πr^2. However, since we are only interested in the outer surface area, we need to subtract the inner surface area from this, giving us A = 4π(r2^2 - r1^2).

Plugging this into our formula for resistance, we get R = (ρ*(r2-r1))/(4π(r2^2 - r1^2)).

Finally, we can substitute the definition of conductivity (σ = 1/ρ) to get our final formula for the resistance of a hollow sphere: R = (r2 - r1)/(4πσr1r2). This formula takes into account the material's conductivity, as well as the dimensions of the spherical shell.

In conclusion, the total resistance of a spherical shell made of material with conductivity σ and inner and outer radii r1 and r2, respectively, is given by the formula R = (r2 - r1)/(4πσr1r2). This formula can be used to calculate the resistance of any hollow spherical shell, as long as the current is flowing radially outward.