2) A solid nonconducting sphere of radius R = 5.6cm has a nonuniform charge distribution or volume
charge density p= (1.41pCm^-3)r/R, where r is the radial distance from the sphere's center.
a) What is the sphere's total charge?
b) Calculate AND sketch a graph of E versus r.
To find the answers to these questions, we will need to integrate the provided charge density function over the sphere's volume and then use the resulting charge distribution to calculate the electric field at each point.
a) To determine the sphere's total charge, we need to integrate the charge density function over its volume. The charge density function p is given by p = (1.41 pCm^-3)r/R, where r is the radial distance and R is the sphere's radius.
To integrate over the volume, we can use the spherical coordinates system. The volume element in spherical coordinates is given by dV = r^2 sin(θ) dr dθ dφ. Since the charge density depends only on the radial distance r, we can integrate over the other variables to simplify the expression.
The integral for the total charge Q is then:
Q = ∫∫∫ p dV
Q = ∫(0 to 2π) ∫(0 to π) ∫(0 to R) (1.41 pCm^-3)r/R r^2 sin(θ) dr dθ dφ
Evaluating this integral will give us the sphere's total charge in the units of coulombs (C).
b) To calculate the electric field (E) as a function of the radial distance (r), we can use Gauss's Law. Gauss's Law states that the electric field through a closed surface is proportional to the net charge enclosed by that surface.
First, we need to determine the electric field at a distance r from the center of the sphere. We can consider a Gaussian surface in the form of a concentric sphere with radius r. The electric field will be radial and have a magnitude E throughout the surface.
Using Gauss's Law, we have:
∮ E ∙ dA = (Qenc) / ε0
Here, ∮ E ∙ dA represents the surface integral of the electric field over the Gaussian surface, Qenc is the net charge enclosed by the Gaussian surface, and ε0 is the electric constant.
For our case, Qenc will depend on the distance r from the center of the sphere, as the charge distribution is nonuniform. Thus, we need to express Qenc as a function of r.
To calculate the electric field at each point, we will differentiate the charge accumulated within a sphere of radius r with respect to r. Then, E can be calculated as E = (dQenc / dA) / ε0, where dQenc is the derivative of Qenc with respect to r and dA is the differential area element on the Gaussian surface.
Using this equation, we can calculate the electric field at each point and plot a graph of E versus r.
Please note that the calculations involved in these problems are quite detailed and require some mathematical skills. It is recommended to use appropriate mathematical software or consult a textbook for further guidance if needed.