electric current - Resistance and potential difference across it

I have this question that has baffled me for hours now. My question is what really happens when charge passes from one point to another in a circuit for eg what happens when it passes over a resistor. I have referred to many Q&A on SE. Here's what I have learned:

1. Whenever there is a battery connected in a circuit we assume that the positive terminal of the battery is at a higher potential then the negative terminal by convention. Also by convention + charge flows from positive to negative terminal .The driving force here is the potential difference.


2. Electrons flow in the opposite direction. The driving force here is the force exerted by the electric field.


3. 
   
   

   Electric field direction is from positive to negative and if a particle travels in its direction , it loses potential energy and vice versa.(since I learned that electric field lines point in the direction of decreasing electric potential)

   
   


4. 
   

   As electrons move they constantly gain potential energy,but they also gain kinetic energy as they are being accelerated by E .So isn't it violating conservation of energy?

   
   


5. For an electron entering a resistor Initial KE+Initial PE=Final KE +Final PE+Heat dissipated. But since Initial KE=Final KE. Therefore Initial PE should be greater than Final PE but according to point 3 this cannot happen.So what is happening?

Also why do we say that current everywhere in a circuit is the same,even before and after entering a resistance?

Answer

1. Whenever there is a battery connected in a circuit we assume that the positive terminal of the battery is at a higher potential then the negative terminal by convention. Also by convention + charge flows from positive to negative terminal .The driving force here is the potential difference.

Correct

2. Electrons flow in the opposite direction. The driving force here is the force exerted by the electric field.

Correct.

But don't be confused, the driving force here is the same as before. The electric potential difference $V$ corresponds to an electric field $E=\frac{dV}{dx}=\frac{V}{L}$ (the change or gradient of the potential is the field), which causes this electric force $F=Eq$ (where $q$ here is the negative electron charge, and thus the force is opposite towards the higher potential).

3. Electric field direction is from positive to negative and if a particle travels in its direction , it loses potential energy and vice versa.(since I learned that electric field lines point in the direction of decreasing electric potential)

Correct, if you to your last line add: "...for a positive charge".

4. As electrons move they constantly gain potential energy,but they also gain kinetic energy as they are being accelerated by E .So isn't it violating conservation of energy?

No. See the comment for point 3. The electrons are as well moving in the direction of decreasing potential energy.

Even though electrons are moving towards higher potential $V$ they are still moving towards lower potential energy, because $U=Vq$. $q$ is negative for electrons and thus the potential energy $U$ here changed sign and is lower than at the other pole.

A note: Remember that electrons moving one way corresponds to positive charge moving the other way. So what causes positive charge to flow is exactly the same mechanism that makes negative charge flow, just looked upon up-side-down if you will.

5. For an electron entering a resistor Initial KE+Initial PE=Final KE +Final PE+Heat dissipated. But since Initial KE=Final KE. Therefore Initial PE should be greater than Final PE but according to point 3 this cannot happen.So what is happening?

You forget here that there is much heat loss in an electric circuit. The complete energy conservation equation says:

$$K_{initial} + U_{initial}=K_{final}+U_{final}+W_{loss}$$

$W_{loss}$ is any energy loss and for a circuit this is heat. Heat generated and lost to the ambient air in resistances in components and wires etc.

Note: The energy that is converted into heat comes from the loss in potential energy - the kinectic energy of the electron remains the same - so $W_{loss}=U_{initial}-U_{final}=-\Delta U$

Also why do we say that current everywhere in a circuit is the same,even before and after entering a resistance?

This is Kirchhoff's current law:

$$\sum i = 0\implies i_{in}=i_{out}$$

It simply states that charge does not accumulate anywhere. All charge entering at any point must also leave at the same rate, so no charge is build up. Current in must equal current out.

This is intuitively logical:

• There cannot leave more charge every second than what enters every second at a point, because where does the charge that leaves come from then. So current out cannot be higher than current in.


• And if more current enters than what leaves, then charge will build up infinitly. Soon you will have a huge amount of charge here. This huge chunk of negative charge will for sure repel any further incoming electrons and stop the current. So if you do have a current running, then you know that no charge is build up anywhere, and what enters must equal what leaves.