Finding the value of the element dq

[warrior_1]

Homework Statement

Ok the question states the following, explain why the element of charge, dq, located within an infinitesimally thin spherical shell or radius r is equal to rho(r)4$$\pi$$(r)^2dr, where dr is the thickness of the shell

Homework Equations

The Attempt at a Solution

ok i know that total charge is equal to charge density multiplied by volume, which is equal to rho*4/3pi*r^2. Hence if i were to find dq, i would have to differentiate with respect to dq/dr and solve for dq... thus dq/dr=rho(r)*dv/dr
where dv/dr=4*pi*r^2, therefore if i solve for dq i should get dq=rho(r)*4*pi*r^2*dr...

ok i have no idea if that was right or not... any hints and also why don't we differentiate rho(r)

 

[anathema]

in the first place?

Hi there,

Your solution is correct! The reason we don't differentiate rho(r) in the first place is because it is already a function of r. The equation for total charge, q, is q = rho(r)*4/3pi*r^3. Since we are only looking at an infinitesimally thin shell, we can ignore the volume term, 4/3pi*r^3, and focus on the charge density, rho(r), which is already a function of r.

Hope this helps! Let me know if you have any other questions.

 

[tomcue]

in this case?I would like to clarify and provide a more thorough explanation for the given problem.

Firstly, the element of charge, dq, refers to a small amount of charge located within an infinitesimally thin spherical shell of radius r. This means that we are considering a small portion of charge within a very thin spherical shell.

The charge density, rho(r), refers to the amount of charge per unit volume at a given distance r from the center of the sphere. In other words, it describes how the charge is distributed within the sphere.

Now, let's consider the infinitesimally thin spherical shell. The thickness of the shell, dr, refers to the thickness of the layer of charge within the shell. As the shell is infinitesimally thin, we can assume that the charge is evenly distributed within the shell. Therefore, the volume of this thin shell can be approximated as the surface area of the shell multiplied by its thickness, which is 4*pi*r^2*dr.

Using the equation for total charge, we can express dq as the product of charge density and volume, which in this case is rho(r)*4*pi*r^2*dr. This is because the charge density, rho(r), is the amount of charge per unit volume, and we have determined the volume of the thin shell to be 4*pi*r^2*dr.

To address the question of why we do not differentiate rho(r) in this case, it is because we are considering a small portion of charge within the infinitesimally thin shell. This means that the charge density, rho(r), is assumed to be constant within this small portion. Therefore, it is not necessary to differentiate it.

In conclusion, the value of the element of charge, dq, located within an infinitesimally thin spherical shell of radius r is equal to rho(r)*4*pi*r^2*dr. This is because dq represents a small portion of charge, rho(r) is the charge density at a given distance r, and the volume of the thin shell is 4*pi*r^2*dr.