# How does energy flow in a circuit?

I read some information on a certain blog saying that energy does not flow inside the wires of a circuit, it propagates in the form of an electromagnetic field outside the wires and straight into the electric components.

My intuition told me that electrons moving inside the wires move in the same way a ball falling from the sky towards the ground moves, therefore gaining kinetic energy along the way. Since a lot of recent textbooks treat the battery as a form of a capacitor which drives electrons to one side of the plate using a certain mechanism similar to that of a conveyer belt( which isn't different from raising a ball to give it potential energy). So which idea is wrong or are both ideas correct?

The energy flows with the electromagnetic wave. As a demonstration, a wire of current $I$ and potential difference $V$, has the common fields

$$|\mathbf{E}| = \frac{V}{L}$$

and

$$|\mathbf{B}| = \frac{\mu_0 I}{2\pi r}.$$

You can derive the power emitted in this wire using the Poynting vector $\mathbf{S}$, which describes the intensity of the electromagnetic wave. Then, the power is

$$P = \oint \mathbf{S} \cdot d\mathbf{a} = \frac{1}{\mu_0}\oint\left(\mathbf{E} \times \mathbf{B}\right)\cdot d\mathbf{a},$$

where $d\mathbf{a}$ is a differential surface area element of the cylindrical wire. You see that

$$P = \frac{1}{\mu_0}\cdot \frac{V}{L} \cdot \frac{\mu_0 I}{2\pi r} \cdot 2\pi rL = IV.$$

Your intuition for the electron's motion is not completely incorrect. Essentially, charges in wires can modeled like incompressible fluid flow. This is done in the Drude-Lorentz model. Essentially the electrons are modeled as an ideal gas, having elastic collisions with the positive ions of the conductor. As such, there is a total drift speed to the electrons, although each one is always accelerating due to the collisions. The drift speed is directly proportional to the electric field.

The important part is that this does not matter. The information is not transmitted in the arrival of moving electrons, it is transmitted in the electromagnetic field. Electronic information travels from wires at nearly the speed of light, even though the electrons in the conductor move quite slowly.

• Thank you, now i have got a good rough idea, but the part i don't get is if the electrons move with the same drift velocity throughout why do some textbooks say that after cycling the circuit, the electrons go to the battery to gain energy again. Isn't their energy constant since their velocity is the same everywhere in the circuit? Sorry i do not possess the link, it's a blog i was on about a year ago.. – Energy Aug 4 '18 at 12:16