5. Write the differential and integral forms of the fundamental postulates of electrostatics in free space. system after each charge is positions.

The differential and integral forms of the fundamental postulates of electrostatics in free space are:<br /><br />Differential Form:<br />1. Gauss's Law: The total electric flux through a closed Gaussian surface is proportional to the total charge enclosed by the surface. Mathematically, it is represented as:<br /><br />∇⋅E = (1/ε₀)∑qenclosed<br /><br />where E is the electric field, ε₀ is the permittivity of free space, and qenclosed is the total charge enclosed by the Gaussian surface.<br /><br />2. Faraday's Law of Induction: The rate of change of the magnetic flux through a closed loop is equal to the negative rate of change of the electric potential around the loop. Mathematically, it is represented as:<br /><br />−∂ΦB/∂t = ∂V/∂t<br /><br />where ΦB is the magnetic flux, V is the electric potential, and t is time.<br /><br />Integral Form:<br />1. Gauss's Law: The total electric flux through a closed Gaussian surface is equal to the total charge enclosed by the surface divided by the permittivity of free space. Mathematically, it is represented as:<br /><br />∫∫∫E⋅dA = (1/ε₀)∑qenclosed<br /><br />where E is the electric field, ε₀ is the permittivity of free space, and qenclosed is the total charge enclosed by the Gaussian surface.<br /><br />2. Faraday's Law of Induction: The induced electric potential around a closed loop is equal to the negative rate of change of the magnetic flux through the loop. Mathematically, it is represented as:<br /><br />∫∫∫(−∂ΦB/∂t)⋅dA = ∫∫∫(−∂V/∂t)⋅dA<br /><br />where ΦB is the magnetic flux, V is the electric potential, and t is time.