Can this physical quantity be simplified into a smaller form?

I have a quantity I call fake current.

$$I' = \frac{\mathrm{d} \oint \nabla \mathcal{E} \cdot \mathrm{d} \vec{A}}{\mathrm{d} t}$$

Current is:

$$I = \frac{\mathrm{d} q}{\mathrm{d} t}$$

Charge is directly proportional to the flux through a closed surface

$$Q = \epsilon_0 \oint \vec{E} \cdot \mathrm{d} \vec{A}$$

The electric field can be derived from voltage:

$$\vec{E} = \nabla V$$

and so

$$I = \frac{\mathrm{d} \oint \nabla V \cdot \mathrm{d} \vec{A}}{\mathrm{d} t}$$

I was wondering if there was a similar thing for magnetism.

$$I' = \frac{\mathrm{d} \oint \nabla \mathcal{E} \cdot \mathrm{d} \vec{A}}{\mathrm{d} t}$$

However, I'm having trouble comprehending this quantity. Is it possible to simplify this relationship mathematically?

• I am not sure but i think a vector potential is required to get the electric field from time varying potential. – hsinghal Jul 22 '16 at 17:47

https://en.wikipedia.org/wiki/Magnetic_monopole You are correct in your intuition that there would be a "Magnetic Current" if monopoles existed. Indeed it is fun to imagine a beautiful symmetry between the Electric Force and Magnetic Force, but alas, nature only gives us half of it.

No it isnt possible to simplify this relationship mathmatically. It takes the same form for "Magnetic Charge" as it does for "Electric Charge". Thats the symmtetry of it.

Although I suppose that if you really wanted to simply your equation, we could invoke Gauss's Law for Magnetism:

$\oint \vec{B} \cdot d\vec{a} = 0$

$\vec{B}=-\vec{\nabla}\mathcal{E}+\vec{\nabla} \times \vec{A}$ , $\vec{A}$ is the vector potential

$\oint (-\vec{\nabla} \mathcal{E} +\vec{\nabla} \times \vec{A})\cdot d\vec{a} = \oint (-\vec{\nabla} \mathcal{E} )\cdot d\vec{a} = \Phi_B$

since the surface integral of a curl is $0$