Gauss law of electricity | Electrostatics | Physics | Khan Academy

The video begins by setting the stage for an exploration of the law of electricity, specifically focusing on the concept of electric flux. The host introduces a hypothetical scenario involving a positively charged sphere and the resulting electric field. The primary question posed is how to calculate the electric flux through the sphere. The explanation delves into the definition of flux as the integral of the electric field (E) dot product with a differential area (dA), taking into account the cosine of the angle (theta) between the field and the area vector. The significance of the angle and the perpendicular component of the area vector to the field is emphasized, leading to the understanding that the flux is calculated using the dot product of E and dA. The recap encourages viewers to revisit previous material if they are not familiar with the concept.

This paragraph continues the discussion on electric flux, focusing on calculating the flux through a sphere with a point charge at its center. The host guides the viewer through the process of determining the angle between the electric field and the differential area vector (dA), concluding that the angle is zero everywhere on the sphere due to the field's alignment with the radius. This simplifies the calculation because cos(0) equals 1. The electric field is identified as constant across the sphere's surface because all points are equidistant from the charge. The integral of dA over the sphere's surface is the total surface area, which is 4πr². Using Coulomb's law, the electric field E is expressed as q/(4πε₀r²), and the total flux is found to be q/ε₀ after simplification. The host also discusses the implications of changing the sphere's size and position, highlighting the independence of flux from the sphere's size due to the inverse square law of the electric field.

The host introduces Gauss's law by exploring the implications of moving the charge off-center and the effect on the electric flux through the sphere. Initially, the derivation becomes more complex due to the non-uniform electric field and varying angle theta. However, the host reveals that the flux remains unchanged, despite the charge's displacement, which is a surprising result attributed to the inverse square law. The concept is further expanded by imagining the electric field as 'bullets' being emitted by the charge, illustrating that the flux, or the number of 'bullets' passing through the sphere, remains constant regardless of the sphere's position or the charge's location. The host also touches on scenarios involving multiple charges and closed surfaces, leading to the general principle that the flux through a closed surface equals the total charge enclosed, divided by the permittivity of free space (ε₀).

The final paragraph encapsulates the essence of Gauss's law in electrostatics. The host reiterates that the law, derived for a simple case of a charge at the center of a sphere, is universally applicable to any closed surface, regardless of its shape or the position of the charge. The law states that the total electric flux through a closed surface is equal to the total charge enclosed by the surface, divided by the permittivity of free space (ε₀). The host emphasizes that charges outside the surface do not affect the flux, and the law holds true even if there are multiple charges or if the surface has an irregular shape, as long as it is closed. The paragraph concludes by highlighting the generality and importance of Gauss's law in understanding electrostatic fields.