Integrating ##\sigma=\chi\int{dA/A}## for a sphere

[MartynaJ]

Homework Statement

Integral of $\sigma=\chi\int{dA/A}$ for a sphere assuming a constant and surface-area independent $\chi$

Relevant Equations

I am trying to integrate $\sigma=\chi\int{dA/A}$ for a sphere. The answer is supposed to be $\sigma(R)=\chi(R^2/R_0^2-1)$

I am trying to integrate $\sigma=\chi\int\frac{dA}{A}$ for a sphere. The answer is supposed to be $\sigma(R)=\chi(R^2/R_0^2-1)$. The answer I keep getting is $\sigma(R)=2\chi ln\frac{R}{R_0}$. I also tried doing it in spherical coordinates, and all I get for the integration of $\int_0^\frac{\pi}{2}\frac{dA}{A}=1$. Not sure where I am going wrong... please help!

 

[pasmith]

It would help if you show us how you obtained your answer.

 

[kuruman]

Homework Statement:: Integral of $\sigma=\chi\int{dA/A}$ for a sphere assuming a constant and surface-area independent $\chi$

Is this all there is to the homework statement? If there is more, please post exactly as given to you.

Also, $\sigma=\chi\int{dA/A}$ is ambiguous. Is it $\sigma=\chi\int{\dfrac{dA}{A}}$ or $\sigma=\chi\dfrac{1}{A}\int{dA}$? In the second interpretation $A$ stands for the area of the sphere, presumably of radius $R_0$. Finally, it would help if you told us whether there is some physical meaning to $\sigma(R).$

 

[MartynaJ]

Is this all there is to the homework statement? If there is more, please post exactly as given to you.

Also, $\sigma=\chi\int{dA/A}$ is ambiguous. Is it $\sigma=\chi\int{\dfrac{dA}{A}}$ or $\sigma=\chi\dfrac{1}{A}\int{dA}$? In the second interpretation $A$ stands for the area of the sphere, presumably of radius $R_0$. Finally, it would help if you told us whether there is some physical meaning to $\sigma(R).$

 

[MartynaJ]

It's $\sigma=\chi\int{\dfrac{dA}{A}}$. Also $\sigma(R)$ is the surface tension and $\chi$ is the stiffness of the spherical bubble (assumed to be constant).

 

[haruspex]

It's $\sigma=\chi\int{\dfrac{dA}{A}}$. Also $\sigma(R)$ is the surface tension and $\chi$ is the stiffness of the spherical bubble (assumed to be constant).

I don't think any of us reading this understand what is meant by the A in the denominator. The dA in the numerator suggests an area element for an integral performed over the surface of a sphere, presumably of radius R, but then the A in the denominator would have to be a function of the chosen element.
From the answer you got, I am guessing you substituted $A=4\pi r^2$ in both places and treated at as an integral dr.

 

[kuruman]

It's $\sigma=\chi\int{\dfrac{dA}{A}}$. Also $\sigma(R)$ is the surface tension and $\chi$ is the stiffness of the spherical bubble (assumed to be constant).

According to the desired answer, $\sigma(R_0)=0.$ Why is the surface tension zero for a specific value of the bubble radius?