The orgin of the superposition principle of electric field

[feynman1]

Isn't the superposition principle of electric field just force being addable? Jackson's electrodynamics says it's based on the premise of linear Maxwell's equations. Which support(s) the superposition principle?

 

[etotheipi]

If something is linear, e.g. Gauss' law, then $$\nabla \cdot (\mathbf{E}_1 + \mathbf{E}_2) = \nabla \cdot \mathbf{E}_1 + \nabla \cdot \mathbf{E}_2$$ So really if you have $\nabla \cdot \mathbf{E}_1 = \frac{\rho_1}{\epsilon_0}$ and $\nabla \cdot \mathbf{E}_2 = \frac{\rho_2}{\epsilon_0}$ then you also have $\nabla \cdot (\mathbf{E}_1 + \mathbf{E}_2) = \frac{\rho_1 + \rho_2}{\epsilon_0}$

It's the same principle that let's you superpose two solutions to the wave equation.

 

[feynman1]

If something is linear, e.g. Gauss' law, then $$\nabla \cdot (\mathbf{E}_1 + \mathbf{E}_2) = \nabla \cdot \mathbf{E}_1 + \nabla \cdot \mathbf{E}_2$$ So really if you have $\nabla \cdot \mathbf{E}_1 = \frac{\rho_1}{\epsilon_0}$ and $\nabla \cdot \mathbf{E}_2 = \frac{\rho_2}{\epsilon_0}$ then you also have $\nabla \cdot (\mathbf{E}_1 + \mathbf{E}_2) = \frac{\rho_1 + \rho_2}{\epsilon_0}$

It's the same principle that let's you superpose two solutions to the wave equation.

Thanks. I think you disagree with force being addable being the basis of superposition of electricity. If Maxwell's equations were nonlinear, forces would still be generally addable yet the vector sum of forces wouldn't be the correct net force, but rather just another force.

 

[A.T.]

Isn't the superposition principle of electric field just force being addable? Jackson's electrodynamics says it's based on the premise of linear Maxwell's equations. Which support(s) the superposition principle?

"Linear" is the technical term for "addable": f (a+b) = f(a) + f(b)

 

[etotheipi]

Thanks. I think you disagree with force being addable being the basis of superposition of electricity. If Maxwell's equations were nonlinear, forces would still be generally addable yet the vector sum of forces wouldn't be the correct net force, but rather just another force.

I'm not sure if I quite follow. The net force is the vector sum of all of the forces, that doesn't directly relate to Maxwell's equations. It's still always true.

You could apply the principle of superposition to forces at a point on an object too. If $\vec{F}_1 = m\vec{a}_1$ and $\vec{F}_2 = m\vec{a}_2$, then $\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 = m(\vec{a}_1 + \vec{a}_2)$.

It's just two different ways of approaching the same thing.

 

[feynman1]

"Linear" is the technical term for "addable": f (a+b) = f(a) + f(b)

Yes. So do you think linear Maxwell is exactly equivalent to net force being equal to the vector sum of forces?