(a) Suppose the electric field (in spherical coordinates) is found to be kraf, for r a where k and a are some constants.
i. Find the volume charge density p everywhere in space.
ii. Use Gauss's Law to find the total charge contained in a sphere of radius R > a, centred at the origin.
iii. Check your answer to (ii) by integrating the charge density found in (i)
To find the volume charge density (p) everywhere in space, we need to first analyze the given electric field (E). The electric field in spherical coordinates is given as E = k/r^a, where k and a are constants.
(i) Finding the volume charge density (p):
We can relate the electric field to the charge density using Gauss's Law. Gauss's Law states that the total electric flux through a closed surface is directly proportional to the charge enclosed by that surface.
Starting with Gauss's Law:
Integral(E * dA) = Q_enclosed / ε₀,
where E is the electric field, dA is the differential area element, Q_enclosed is the charge enclosed within the surface, and ε₀ is the vacuum permittivity.
Now, consider a closed spherical surface of radius r centered at the origin. The electric field is spherically symmetric, and its magnitude only depends on the radial distance r from the origin. Hence, the electric field will be constant on the surface of the sphere, and its direction will be radially outward. Thus, the electric field at every point on the surface is E = k/R^a, where R is the radius of the sphere.
Now, we can calculate the electric flux through this spherical surface:
Flux = E * 4πR^2,
where R is the radius of the sphere.
Applying Gauss's Law, we have:
Flux = Q_enclosed / ε₀.
Substituting the value of flux and rearranging the equation, we get:
Q_enclosed = E * 4πR^2 * ε₀.
Now, let's find the charge density (p) using the formula:
p = Q_enclosed / V,
where V is the volume enclosed by the sphere.
Considering a spherical volume element, dV, of radius r and thickness dr, the volume element is given by:
dV = 4πr^2 * dr.
As we want to find the volume charge density p everywhere in space, we'll consider the entire space. Thus, the volume V will be the volume for r ≥ a.
Integrating the volume charge density over the volume V:
∫(p dV) = ∫( Q_enclosed / V * 4πr^2 * dr),
where the integration is taken from r = a to infinity.
(ii) Using Gauss's Law to find the total charge contained in a sphere of radius R > a, centered at the origin:
As derived in step (i), the total charge enclosed within a closed surface of radius R is given by:
Q_enclosed = E * 4πR^2 * ε₀,
where E = k/R^a.
Substituting the value of E, we get:
Q_enclosed = (k/R^a) * 4πR^2 * ε₀.
(iii) Checking the answer to (ii) by integrating the charge density found in (i):
As found in step (i), p = Q_enclosed / V * 4πr^2.
To check the answer obtained in step (ii), we can integrate the volume charge density over the volume V:
∫ (p dV) = ∫ ( Q_enclosed / V * 4πr^2 * dr),
where the integration is taken from r = a to R.
Evaluate the integral and compare the result to the answer obtained in step (ii). If they are the same, then the answer is correct. If they differ, recheck the calculations and ensure no mistakes were made.