maxwell equation in 1d
Maxwell's first law in differential form states that $$ \triangledown \cdot E = \frac{\rho}{\epsilon_{o}} $$ . In case of 1d can we say that $$\rho = \lambda$$
where $$\lambda$$ is the linear charge density of something?
1 answer
Yes, that would the obvious interpretation of that equation in one dimension. Note also that in that case, the divergence also reduces to the ordinary derivative.
In other words, in one dimension, the electric field is constant wherever there is no charge, and if we additionally demand that the electric field vanishes at infinity, it follows that the total charge of the one-dimensional universe is zero, and the electric field is completely determined by the difference between the charges on both sides of you, regardless of distance.
Indeed, this is a good approximation to what happens with a plate capacitor if effects at the boundary of the capacitor are neglected.
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