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Ohm is defined as

"a resistance between two points of a conductor when a constant potential difference of 1.0 volt, applied to these points, produces in the conductor a current of 1.0 ampere, the conductor not being the seat of any electromotive force"

according to this wikipedia page. However, I still don't understand what exactly is resistance and ohm. Does resistance decrease the amount of electrons passing through a conductor per unit time or does it decrease the energy that a single electron carries or none of them? I know what Ohm law is and I also know the mathematical background of resistance but I can not understand the logic behind it.

For example, when we talk about a conductor that should be insulated by an electrical insulator we have to know how many ohms that is needed to determine the size of the insulator for a good insulation. But I would like to know how it is determined. Thus, I would be pleased if someone could explain the logical background of resistance and ohm. Thanks.

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2 Answers 2

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An intuitive tool to understand resistivity in the classical context, is Drude model.

Consider a 3 Dimensional lattice, of stationary obstacles, that are separated by a characteristic length $\lambda$.

Now consider a charged particle, operating under a potential difference that creates a constant E-field in this 3D lattice, let's say in the $\hat{x}$ direction.

Now, without the 3D lattice, you would expect the charged particle to be accelerated in a constant fashion, since: $$ \vec{F}\equiv e\vec{E} $$

But that is not the case here, what actually happens, is the particle keeps bumping into these obstacles, and the velocity component, that was generated strictly in the $\hat{x}$ direction: $v_x$ is now "thermalized" i.e. it gets deflected to a random direction that, over all averages to zero thus: $$ <v_T>=0 $$ That is true because of the random nature of this velocity.

This means there is only a finite length in which the particle can accumulate velocity in the $\hat{x}$ direction so - here is a complete derivation: $$ V_x = a*\tau = \frac{e\nabla\Phi}{m_p}\cdot \tau = \frac{eE_x}{m_p}\cdot\tau $$ (up to a numeric constant).

Let's find $\tau$: $$ \tau\propto \frac{"x"}{"v"}=\frac{\lambda}{V} $$ Where as here we take ALL of the velocity, as is, and not the averaged component: $$ \Rightarrow \tau\propto\frac{\lambda}{V_T + V_x} $$ Where $|V_T|\gg |V_x|$ so we can just go ahead and say: $$ \Rightarrow \tau\propto\frac{\lambda}{V_T} $$ plugging these in: $$ v_x\propto\frac{eE_x \lambda}{m_p V_T} $$

Now we want to consider the current so we take the amount of charged particels that go through the whole resistor in a second which is just: $$ I=qAv_x $$ Where $q$ is the charge of the particle, and we get: (changed $e\rightarrow q$) $$ I\propto \frac{qE_x \lambda }{m_p V_T}\cdot\frac{qA\cdot l}{l} $$ Now we bunch these together nicely: $$ V=E_x l $$ $$ \rho = \frac{m_p V_T}{q^2\lambda} $$ And we get Ohm's law: $$ I=\frac{V}{R} $$ Where $$ R=\rho\cdot \frac{l}{A} $$

These should give you a sense for resistivity, when the Temperature of the resistor increases, the resistivity shoots up. similarly when the spacing between obstacles increase, the resistivity goes down.

Another way one can think of it, is by looking at $$ F=m\cdot a $$ So, now look at: $$ V=I\cdot R $$ In classical mech, you can think of the mass as, how hard it is to motivate an object to move, the analogue for electricity in this case will be, $R$ states how hard it is, given potential $V$, to cause a movement of electrons through the resistor, to form a current.

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  • $\begingroup$ Of course this flies out the window in high enough temperatures, and/or dealing with quantum resistivity... This is ONLY to give some classical intuition. $\endgroup$
    – BeastRaban
    Oct 7, 2014 at 21:18
  • $\begingroup$ Thanks for your answer. I got stuck on what $V_t$ and $V_x$ are. Can you please explain it to me? $\endgroup$
    – Starior
    Oct 8, 2014 at 14:59
  • $\begingroup$ Sure! Basically the premise is that $V_T$ is the velocity in a random direction. This is the so called "thermal" velocity. The name is given to this velocity since, essentially this velocity constitutes "heat" by the relation $T\propto\frac{mV^2}{2}$. $V_x$ is the velocity component in the x direction that originates in the potential difference. You may also call $V_x=V_D$ as in DRIFT velocity. These youtube clips might shed some light: youtube.com/watch?v=KgbqPKZU5IA youtube.com/watch?v=agI6rclA_vQ $\endgroup$
    – BeastRaban
    Oct 9, 2014 at 16:04
  • $\begingroup$ Actualy $\langle V_X \rangle \equiv V_D$, while $\langle V_T \rangle=0$ since the thermal velocity is random in direction it cancels out when averaged over time and/or particles and/or interactions. $\endgroup$
    – BeastRaban
    Oct 9, 2014 at 16:19
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An ohm is one volt per amp.

That is, a one ohm (ideal, linear) resistor requires 1 V in order for 1 A to pass through it. A 2 $\Omega$ resistor requires 2 V in order for 1 A to flow. Conversely, if 1 A is forced to flow through a 1 $\Omega$ resistor, 1 V will be developed across the resistor.

Does resistance decrease the amount of electrons passing through a conductor per unit time or does it decrease the energy that a single electron carries or none of them?

The resistance of a wire restricts the net flow of electrons for a given voltage applied between the two ends of the wire. But the electrons that flow in one end of the wire are not generally the same ones that flow out the other end, so there's no use talking about the energy of any particular electron flowing through the wire.

For example, when we talk about a conductor that should be insulated by an electrical insulator we have to know how many ohms that is needed to determine the size of the insulator for a good insulation.

This is not correct in practice. While its true we want to limit the current that flows through the insulation, other considerations are more critical to the choice of insulation.

Insulation is designed for the maximum voltage that might be applied to the conductor relative to surrounding objects (for example, the earth itself, or an earthed object). The material and thickness must be chosen so that the electric field between the wire and the objects around it isn't more than the breakdown voltage of the insulation.

The resistance of the wire itself causes a potential difference between the two ends of the wire. But even if this difference is small (as it should be), the potential relative to nearby objects could still be very large. The resistance of the wire might contribute to the wire heating up in use, and the insulation must be designed to withstand whatever temperature the wire might reach without catching fire or melting.

But neither the breakdown voltage nor the thermal limits of the insulation are particularly strongly related to its resistance.

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