Question

Derive coulomb's law from maxwell's equations

Answer 1

k●14●151●324
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The exact derivation goes as follows. You start from Gauss' Law, integrate on both sides over some volume V:

∇⃗ div⋅E⃗ =1ϵ0ρ/∭Vd3r⃗
∇→div⋅E→=1ϵ0ρ/∭Vd3r→
Then switch to integration over a closed surface, and also note that total charge inside this volume is Q:
∭V∇⃗ div⋅E⃗ d3r⃗ =∮E⃗ ⋅dσ⃗ =∭V1ϵ0ρd3r⃗ =Qϵ0
∭V∇→div⋅E→d3r→=∮E→⋅dσ→=∭V1ϵ0ρd3r→=Qϵ0
Now you need to note that the volume of integration is quite arbitrary and so is the surface, so we will use a sphere. You can describe the integral over a sphere using:
Qϵ0=∮E⃗ ⋅dσ⃗ =∫ϕ=0ϕ=2π∫θ=0θ=πEn⃗ ^⋅n⃗ ^RdϕRdθ=4πR2E/14πR2
Qϵ0=∮E→⋅dσ→=∫ϕ=0ϕ=2π∫θ=0θ=πEn→^⋅n→^RdϕRdθ=4πR2E/14πR2
And so you obtain:
E=Q4πϵ0R2
E=Q4πϵ0R2
It should be:
E⃗ =Q4πϵ0R2r^
E→=Q4πϵ0R2r^
But I lost the normal vector along the way (I hope that someone can correct this and edit this post).

Now you use the Lorentz Force law (where B⃗ =0⃗ B→=0→):
F⃗ lor=qE⃗ +qV⃗ ×B⃗ =qQ4πϵ0R2r^
F→lor=qE→+qV→×B→=qQ4πϵ0R2r^
And so you obtain the Coulomb force law

HOPE THIS WILL HELP U!! ◆◆◆
PLS MARK AS BRAINLEST...

Answer 2

Answer:

Maxwell's equations are a set of four differential equations that form the theoretical basis for describing classical electromagnetism: Gauss's law: Electric charges produce an electric field. The electric flux across a closed surface is proportional to the charge enclosed.