Question

Figure 23-24 shows, in cross section, two Gaussian spheres and two Gaussian cubes that are centered on a positively charged particle. (a) Rank the net flux through the four Gaussian surfaces, greatest first. (b) Rank the magnitudes of the electric fields on the surfaces, greatest first, and indicate whether the magnitudes are uniform or variable along each surface.

Short answer

1. The rank of the net flux through the four Gaussian surfaces is $a=b=c=d$.
2. The rank of the magnitudes of the electric fields on the surfaces is $a\left(\text{constant}\right)>b\left(\text{variable}\right)>c\left(\text{constant}\right)>d\left(\text{constant}\right)$.

Step-by-step solution

The given data:

Figure 23-24 showing the cross-section of two Gaussian spheres and two Gaussian cubes that are centered on a positively charged particle is given.

Understanding the concept of Gaussian surface

The net flux through any given surface can be given by the charge enclosed within any Gaussian surface. Now, the electric field value depends only on the area for a given constant enclosed charge within the given Gaussian surfaces.

Formulae:

The electric flux through any closed surface,

${\varphi }_E=\frac{q_{enc}}{{\epsilon }_0}$ ….. (i)

The electric field at any point of a Gaussian surface,

$E=\frac{q_{enc}}{A{\epsilon }_0}$…..(ii)

Here,E is the electric field, $q_{enc}$ is the enclosed charge, A is the area, and ${\epsilon }_0$ is the permittivity of the free space.

(a) Calculation of the rank of the net flux through the four surfaces:

For the given spherical and cubical surfaces, the charge enclosed is

$q_{enc}=q$

From equation (i) provided all the other terms being constant, the net flux through all the given surfaces is constant and thus is given using equation (i) as follows:

${\varphi }_E=\frac{q}{{\epsilon }_0}$

Hence, the rank of the net electric flux through all the surfaces is $a=b=c=d$.

(b) Calculation of the rank of the surfaces according to their electric fields:

For the given spherical and cubical surfaces, the charge enclosed is

$q_{enc}=q$

Thus, the electric field with charge value constant only depends on the area of the Gaussian surface as,

$E\alpha \frac{1}{A}$

Now, the rank of the Gaussian surfaces according to their surface area can be given as,

$a<b<c<d$

Here, the electric field of the spherical surfaces a and c can be given by,

(Surface area of sphere,$A=4\pi R^2$)

$E=\frac{q}{4\pi R^2{\in }_0}$

So, the distance of each point on the spherical surface is equal, thus the electric field is constant.

Now, as the distance of every point is not equal at the square surface of the cubical Gaussian surfaces, thus the electric field is not same.

Thus, the electric field for cube surfaces b and d is variable.

Hence, the rank of the surfaces according to fields is,

$a\left(\text{constant}\right)>b\left(\text{variable}\right)>c\left(\text{constant}\right)>d\left(\text{constant}\right)$