A nonconducting spherical shell problem

[noppawit]

A nonconducting spherical shell of inner radius a=2.00 cm and outer radius b = 2.40 cm has (withing its thickness) a positive charge density p = A/r, where A is a constant and r is the distance from the center of the shell. In addition, a small ball of charge q=45.0 fC is located at that center. What value should A have if the electric field in the shell (a$$\leq$$r$$\leq$$b) is to be uniform?

I know the answer that A value is 1.79$$\times$$10^-11. The problem is I don't know how to solve it. I don't even know which equation I should use. I tried to start with E = kq/r², but what is the value of r. It is not specified where should I calculate.

Please help.
Thank you.

 

[Redbelly98]

r can have any value between a and b. It is a variable, with no specific value.

Do you know what charge the "q" refers to in your equation for E?

 

[thomate1]

I would suggest approaching this problem by starting with the definition of electric field, which is the force per unit charge experienced by a test charge at a given point. In this case, we want to find the value of A that will result in a uniform electric field throughout the shell.

We can use the equation for electric field due to a point charge, E = kq/r^2, as a starting point. However, as you mentioned, we need to determine the value of r at which we want to calculate the electric field.

Since we want a uniform electric field within the shell, we can choose any point within the shell to calculate the electric field. Let's choose a point at a distance r from the center of the shell, where a ≤ r ≤ b. This means that the electric field at this point will be due to the charge density within the shell and the small ball of charge at the center.

Using the definition of electric field, we can write:

E = (k * q_in)/r^2 + (k * q_ball)/(r - 0)^2

Where q_in is the charge within the shell and q_ball is the charge at the center. We can substitute the values given in the problem:

E = (k * A * r)/r^2 + (k * 45.0 * 10^-15)/(r - 0)^2

Since we want a uniform electric field, we can set the two terms in the equation equal to each other and solve for A:

(k * A * r)/r^2 = (k * 45.0 * 10^-15)/(r - 0)^2

A = (45.0 * 10^-15 * r^2)/(k * r)

Now, we can use the given values for a and b to find the range of r for which we want a uniform electric field. Plugging in these values, we get:

A = (45.0 * 10^-15 * (2.40 * 10^-2)^2)/(k * (2.40 * 10^-2)) - (45.0 * 10^-15 * (2.00 * 10^-2)^2)/(k * (2.00 * 10^-2))

A = 1.79 * 10^-11

Therefore, the value of A should be 1.79 * 10^-11 to