91Ó°ÊÓ

Gauss's Law is a fundamental principle in the study of electric fields and flux. It states that the net electric flux through any closed surface, also known as a Gaussian surface, is directly proportional to the net charge enclosed within that surface. Mathematically, it is expressed as: \[ \Phi_{E} = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \] where \( \Phi_{E} \) is the electric flux, \( \mathbf{E} \) is the electric field, \( d\mathbf{A} \) is the differential area vector, \( Q_{\text{enc}} \) is the enclosed charge, and \( \varepsilon_0 \) is the permittivity of free space.
Gauss's Law is particularly useful for calculating electric fields when dealing with symmetrical charge distributions, such as spherical or cylindrical distributions. In the context of this exercise, even though the electric field strength differs at different points on the sphere, the net electric flux is zero because the sphere does not enclose any charge. This concept underscores how the distribution of electric field lines over the surface maintains balance, compensating for areas with higher intensity by covering wider regions at lower intensity.

Coulomb's Law describes the electrostatic interaction between electric charges. This law states that the magnitude of the electric force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The equation is given by: \[ F = k \frac{|q_1 q_2|}{r^2} \] where \( F \) is the magnitude of the force between the charges, \( k \) is Coulomb's constant \((8.99 \times 10^9 \text{ Nm}^2/\text{C}^2)\), \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.
In the provided exercise, this law helps us determine the electric field magnitude at different locations relative to a point charge. By applying Coulomb's Law, we calculate that a point at \( x = 3.00 \text{ m} \) experiences a stronger electric field compared to a point at \( x = -3.00 \text{ m} \), due to the closer proximity to the point charge. This difference illustrates how electric fields diminish rapidly with distance from a source charge.

Net electric flux refers to the sum of electric field lines passing through a closed surface, which provides insights into the presence and distribution of electric charges. It is computed by the surface integral of the electric field over a closed surface: \[ \Phi_{E} = \oint \mathbf{E} \cdot d\mathbf{A} \] The net electric flux through a closed surface is zero if there are no charges enclosed by that surface.
In the exercise, the sphere centered at the origin does not enclose the external point charge, even though electric field lines are present across the sphere's surface. On the side closer to the charge (at \( x = 3.00 \text{ m} \)), the field lines are denser, indicating a stronger electric field. However, these lines spread out as they pass through the sphere, balancing out the flux. The total flux remains zero because the field lines that enter the sphere eventually exit, maintaining equality between entering and exiting flux.